TIME-OPTIMAL CONTROL OF WHEELED MOBILE ROBOTS

Sergio A. Alvarez and John H. Maddocks

Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD 20742

Abstract. We consider the problem of guiding a general planar wheeled mobile robot (WMR) between given initial and final dynamic states in minimum time. We derive Newton’s equations of motion for such a WMR, expressing balance between inertial forces due to acceleration and frictional forces arising from the contact points between the wheels and the supporting surface. We formulate the minimum time problem as an optimal control problem for the WMR dynamical equations with the wheel velocities as controls. An appropriate version of the Pontryagin maximum principle is used to obtain closed-form expressions for the family of optimal controls. The generation of time optimal trajectories for a given WMR is thereby reduced to a two-point boundary value problem for a system of twelve first order equations involving six parameters. Examples are given for specific unicycle and tricycle WMR designs.

§0: Introduction

There is an extensive literature concerning trajectory generation for autonomous mobile robots (cf. e.g. Cox and Wilfong, 1990?? and references therein). The primary focus of the majority of prior works is the existence of paths in the presence of many obstacles, and the efficient algorithmic generation of feasible trajectories. However there has been comparatively little consideration given to the actual physical dynamics according to Newton’s laws of motion. Here we shall consider these physical dynamics for a wheeled mobile robot (or WMR) that rides over a horizontal plane on a system of wheels. The interaction of the wheels with the supporting plane is modelled by Coulomb’s laws of friction. We derive the system of six first-order ordinary differential equations which express balance between the frictional forces and the angular and linear acceleration of the robot undergoing planar rigid body motion. The relative velocities of the wheels with respect to the robot frame appear in the equations of motion as time dependent control functions. Accordingly we can and do consider the optimal control problem of generating a minimal time trajectory between prescribed initial and final states of the robot. We remark that according to Coulomb’s laws the frictional forces are bounded. Consequently accelerations of the robot are bounded, and we can therefore anticipate

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the well-posedness of the time-optimal control problem. Our main interest lies in the difficulties inherent to the dynamics, and we therefore assume that there are no external constraints restricting the robot motion, i.e. we assume the absence of any collisions.

Problems involving the generation of continuous optimal trajectories of wheeled mobile robots have been considered previously, but the works known to us have been restricted to cost functionals which depend only on the geometry of the trajectory. Consequently all time-dependent phenomena, and in particular inertial effects, are ignored, see e.g. Nelson and Cox (1988), Nelson (1989), Kanayama and Hartman (1989), and Jacobs and Canny (1989). In contrast to the above papers, our modelling takes into account the dynamics and uses geometrically exact rolling conditions for all of the wheels. We are able to apply a version of the Pontryagin maximum principle to obtain, for a general class of WMRs, the family of optimal controls in closed form, thereby reducing the generation of time-optimal trajectories to the solution of a two-point boundary value problem for a system of twelve first-order differential equations. We explicitly construct optimal trajectories for specific unicycle and tricycle WMR designs. Some optimal trajectories correspond to bang-bang acceleration. Interestingly the optimal trajectories we construct also include certain of the “cubic spirals” found by Kanayama and Hartman (1989) as minimizers for a solely geometric cost functional which they call the “integral square jerk”.

As previously remarked we assume that the WMR is moving over a planar surface in the absence of any external objects. In contrast Shiller and Gwo (1991) allow curved terrains and obstacles in their analysis of time-optimal trajectories of a 4-wheeled vehicle. However their model assumes much simplified kinematics and dynamics. Moreover they are concerned mainly with obtaining numerical results, while we concentrate on analytical issues which may lead to an increased understanding of the mathematical structure of the optimal controls and trajectories. In turn, this increased analytical insight should prove useful in designing efficient computational methods for finding time optimal trajectories.

The structure of the paper is as follows. The first section describes our model for the full inertial dynamics of WMRs. The model builds upon the kinematics and quasi-static dynamics described in Alexander and Maddocks (1989). We show how to obtain the Coulomb frictional forces on the wheels in terms of the wheel angular velocity functions, and discuss rolling and sliding of wheels. In the second section we formulate the minimum trajectory time problem as an optimal control problem for the WMR dynamical system with the frictional forces acting at sliding wheels as controls, and with equality side constraints associated with wheels undergoing ideal rolling. The first order necessary conditions for optimality are found by applying a version of the Pontryagin maximum principle for problems with inequality constraints given in Neustadt (1976). The family of extremal controls can be found in closed form, and the minimum-time problem is therefore reduced to a two-point boundary value problem in the original and adjoint variables. Solutions of the optimality conditions for specific WMR designs are considered in the third section. In particular a subclass of the class of “cubic spirals” of Kanayama and Hartman (1989) mentioned above is shown here to correspond to time-minimizing trajectories in certain special circumstances. Other optimal trajectories for a tricycle WMR design are obtained numerically. Our results are summarized and discussed in the concluding section, and some suggestions for further research are made.

1. WMR Dynamics

We regard a wheeled mobile robot (WMR) as a planar rigid body supported by a finite number of massless rigid circular disks (wheels), each of which is attached to the WMR at a fixed point (the axle point) and lies in a possibly time-dependent plane which is normal to the horizontal supporting plane. Each wheel revolves in the robot body about the horizontal line normal to the plane of the wheel and passing through the corresponding axle point. Wheels are classified as either steered or unsteered, and simultaneously as either driven or undriven. Steered wheels may be oriented in any direction with respect to the robot body, while unsteered wheels have a fixed body direction. For driven wheels, the torque about the axle of the wheel may be specified arbitrarily, whereas for undriven wheels the torque must vanish identically. Each wheel is assumed to exert a force on the WMR, due to contact with the supporting surface, in accordance with Coulomb’s law of friction. Both ideal rolling and slippage of the wheels are considered.

In this section of the paper, after a discussion of the basic kinematics of rolling for WMRs, which builds upon that in Alexander and Maddocks (1989), we give Newton’s equations of motion for the WMR described above, in the form of a system of ordinary differential equations with the normalized slippage velocity vectors of the wheels as controls, and show how to obtain, when possible, the frictional force terms which appear in the equations, from knowledge of the angular velocity functions of the wheels. The latter is trivial for slipping wheels, for which Coulomb’s law completely determines the force in terms of the slippage velocity vector, but difficulties are encountered in the case of rolling wheels. The indeterminacy in Coulomb’s law in the rolling case may be reduced by using the WMR dynamics, yielding in certain cases the forces as functions of the dynamic state of the WMR, together with the angular velocities of the wheels and their time derivatives.

Notation.

m: total mass of WMR

CM: center of mass of WMR

I: total moment of inertia of WMR w.r.t. vertical through CM

x(t) ∈ ℝ^{2}: position of CM at time t (w.r.t. ground reference frame)

v(t) ∈ ℝ^{2}: velocity of CM at time t, v(t) := ẋ(t)

θ(t) ∈ ℝ: angular position of WMR body at time t (w.r.t. fixed reference direction in ground frame)

Ω(t) ∈ ℝ: angular velocity of WMR about vertical through CM, Ω(t) := θ̇(t) 1, ..., N: wheels x_{i}(t) ∈ ℝ^{2}: ith axle-point vector a_{i}(t) ∈ ℝ^{2}: angular velocity vector of ith wheel u_{i}(t) ∈ ℝ^{2}: unit slippage velocity vector of ith wheel F_{i}(t) ∈ ℝ^{2}: frictional force on ith wheel at time t r_{i} ∈ ℝ_{+}: radius of ith wheel α_{i} ∈ ℝ_{+}: coefficient of Coulomb friction for contact point between ith wheel and

supporting plane R(ϕ) ∈ SO(2): rotation of ℝ^{2} through the angle ϕ \underline{z} ∈ ℝ^{2}: coordinates of arbitrary vector variable z as seen in the “body frame”,

which is rotated through the angle θ(t) w.r.t. the reference frame ẑ ∈ ℝ^{2}: unit vector in direction z, i.e. ẑ:= z/|z|

Ground frame vs. body frame. It will be convenient to express certain relations in a frame which is rotated through the angle θ(t) with respect to the fixed reference direction, through which the WMR body itself is rotated through at time t. We adopt the convention that if z(t) ∈ ℝ^{2} is any vector variable in the ground reference frame, then \underline{z}(t) ∈ ℝ^{2} denotes the coordinates of z(t) in the rotated “body frame”. Thus, we have

z(t) = R(θ(t))\underline{z}(t)

Differentiation yields

˙ (\underline{z})(t) = \underline{(ż)}(t) − Ω(t)k̂ × \underline{z}(t)

(notice the placing of the parentheses; the left-hand side is the derivative of the body coordinate version of z, while the first term on the right-hand side is the body coordinate version of the derivative of z)

1.1 Kinematics

This section contains the basic kinematics of WMRs. The discussion is based on Alexander and Maddocks (1989).

Types of wheels. We will consider four different wheel types, as follows. Suppose the angular velocity of the ith wheel is written in polar form

a_{i}(t) = |a_{i}(t)|â_{i}(t)

Negative values of the “magnitude” (rotation rate) |a_{i}(t)| are allowed. We say that the wheel is driven iff |a_{i}(·)| may be chosen to be any piecewise continuous real-valued function of time. The wheel is steered iff â_{i}(·) may be chosen to be any piecewise continuous 2-vector-valued function of time. It is undriven iff the torque about its axle vanishes for all t, and unsteered iff the body frame expression ˆ \underline{a}_{i}(t) is constant in time. Thus, a given wheel may be both driven and steered, or driven and unsteered, or undriven and steered, or undriven and unsteered.

Rolling.

The slippage velocity for the i-th wheel at time t, i.e. the relative velocity of the wheel with respect to the supporting surface, at the point of contact, at time t, equals

v(t) + k̂ × (Ω(t)x_{i}(t) + a_{i}(t))

Thus, the ith wheel is rolling at time t iff the following rolling condition holds:

(RC_{i}(t)) : v(t) + k̂ × (Ω(t)x_{i}(t) + a_{i}(t)) = 0

In the body frame, the rolling condition becomes

(\underline{RC}_{i}(t)) : \underline{v}(t) + k̂ × (Ω(t)\underline{x}_{i}(t) + \underline{a}_{i}(t)) = 0

If the i-th wheel is undriven, then the slippage velocity must be parallel to the axle direction \underline{â}_{i} at all times, since any nonzero slippage velocity in the tangential direction k̂ × \underline{â}_{i} would produce a nonzero torque about its axle, according to Coulomb’s dynamic law of friction (see below). The rotation rate |\underline{a}_{i}| automatically adjusts itself to make the tangential component of the slippage velocity vanish. Thus, the i-th wheel will roll iff the normal component (i.e., in the direction \underline{â}_{i}) of the slippage velocity is zero. In other words, the rolling condition for an undriven wheel reduces to

(\underline{RC}_{i}^{ud}(t)) : ⟨\underline{v}(t) + Ω(t)k̂ × \underline{x}_{i}(t), \underline{â}_{i}(t)⟩ = 0

Undriven wheels mounted on the same axle yield the same rolling constraint. In effect, assume wheels i,j are undriven and are placed on a single axle. Then the vector \underline{x}_{i} − \underline{x}_{j} is parallel to the common axle direction \underline{â}, and it follows that the difference between the left sides of the rolling constraints (\underline{RC}_{i}^{ud}), (\underline{RC}_{i}^{ud}), vanishes identically. This argument shows that the same conclusion holds even if wheels i,j are allowed to roll independently, as long as the line connecting their axle points is parallel to their common axle direction.

Inverse Kinematics for Ideal Rolling. The rolling conditions (\underline{RC}_{i}) allow us to determine the unique axle velocities \underline{a}_{i} giving rise to a specified rolling state trajectory (\underline{v}, Ω), namely:

(IK)  \underline{a}_{i} = k̂ × \underline{v} − Ω\underline{x}_{i}

Here, we assume that the resulting axle vectors \underline{a}_{i} are consistent with the given wheel types.

Direct Kinematics for Ideal Rolling. We now wish to address the following problem: given the angular velocities \underline{a}_{i}, find the corresponding state trajectory (\underline{v}, Ω), assuming all wheels are undergoing ideal rolling.

Each rolling condition (\underline{RC}_{i}) defines an affine subspace of (\underline{v}, Ω) 3-space. If the i-th wheel is driven, then this subspace is, according to (\underline{RC}_{i}^{d}), a straight line, dependent upon the parameter \underline{a}_{i}; this line has directing vector −k̂ × \underline{x}_{i} + Ω̂ (where Ω̂ is the unit vector in the direction of the positive Ω axis), and it intersects the \underline{v}-plane at the point \underline{v} = −k̂ × \underline{a}_{i}. If the i-th wheel is undriven, then the associated rolling condition (\underline{RC}_{i}^{ud}) defines a plane through the origin of (\underline{v}, Ω) space, i.e. a 2-dimensional linear subspace of (\underline{v}, Ω) space; since the rolling condition ⟨\underline{v}(t) + Ω(t)k̂ × \underline{x}_{i}(t), \underline{â}_{i}(t)⟩ = 0 holds iff the left side of this equation is a scalar multiple of k̂ × \underline{a}_{i}, we see that the plane is the union of parallel lines, with common directing vector −k̂ × \underline{x}_{i} + Ω̂ , each intersecting the \underline{v} plane at a different point, of the form \underline{v} = −ck̂ × \underline{a}_{i}, for c ranging over all real numbers.

In either case, the set of admissible states (\underline{v}, Ω) is infinite, so at least two wheels are needed to specify a unique state (\underline{v}, Ω). Furthermore, at least one wheel must be driven if it is desired to specify nontrivial states. This is clear intuitively, and may also be understood geometrically in the present context, as corresponding to the fact that any combination of undriven wheels leads to an intersection of linear subspaces and thus that the only state which may be uniquely determined by such wheels is the origin of (\underline{v}, Ω) space, i.e. the zero state. We will now calculate the state (\underline{v}, Ω) determined by two wheels, i and j, i ≠ j. It suffices to consider the case in which at least one of the wheels is driven.

First assume that both wheels are driven. Then the associated lines cannot be parallel, since their directing vectors coincide only if \underline{x}_{i} = \underline{x}_{j}. Thus, the lines are either disjoint (when the angular velocities \underline{a}_{i}, \underline{a}_{j} are mutually incompatible), or they intersect at exactly one point (in this case, (\underline{v}, Ω) is uniquely determined).

From inverse kinematics, if wheels i, j are rolling, then

\underline{a}_{i} = k̂ × \underline{v} − Ω\underline{x}_{i}, \underline{a}_{j} = k̂ × \underline{v} − Ω\underline{x}_{j}

It follows that lines i, j intersect iff (rolling compatibility condition)

\underline{a}_{j} − \underline{a}_{i} = Ω(\underline{x}_{i} − \underline{x}_{j})

The state is then uniquely determined:

Ω = \frac{⟨\underline{a}_{j} − \underline{a}_{i}, \underline{x}_{i} − \underline{x}_{j}⟩}{|\underline{x}_{i} − \underline{x}_{j}|^{2}}

and, from the rolling conditions,

\underline{v} = −k̂ × (\underline{a}_{i} + Ω\underline{x}_{i}) = −k̂ × (\underline{a}_{i} + \frac{⟨\underline{a}_{j} − \underline{a}_{i}, \underline{x}_{i} − \underline{x}_{j}⟩}{|\underline{x}_{i} − \underline{x}_{j}|^{2}} \underline{x}_{i})

This solves the direct kinematics problem in the case in which both wheels i,j are driven.

Now assume the i-th wheel is driven and the j-th wheel is undriven. We will show that the compatibility condition for wheels i,j allows us to uniquely determine the rotation rate |\underline{a}_{j}| of the j-th wheel, in terms of the direction vector \underline{â}_{j} (which is fixed if the j-th wheel is unsteered and variable otherwise, but is known in either case) and of the known axle control \underline{a}_{i} for the driven wheel. This allows us also to express the state (\underline{v}, Ω) of the WMR in terms of the known quantities \underline{a}_{i}, \underline{â}_{j}. Geometrically, in (\underline{v}, Ω) space, this corresponds to finding the intersection of a straight line, given by the rolling condition for the i-th wheel, and a plane, given by the rolling condition for the j-th wheel.

We work in the orthonormal frame \underline{â}_{j}, k̂×\underline{â}_{j} associated with the undriven wheel. We write

\underline{a}_{j} = |\underline{a}_{j}|\underline{â}_{j}

The rolling compatibility condition on wheels i,j becomes

|\underline{a}_{j}|\underline{â}_{j} − \underline{a}_{i} = Ω(\underline{x}_{i} − \underline{x}_{j})

Projection of this equation onto the tangential direction k̂ × \underline{a}_{j} gives us the angular velocity of the WMR:

Ω = −\frac{⟨\underline{a}_{i}, k̂ × \underline{â}_{j}⟩}{⟨\underline{x}_{i} − \underline{x}_{j}, k̂ × \underline{â}_{j}⟩}

Using the rolling condition for the i-th wheel, we find

\underline{v} = −k̂ × (\underline{a}_{i} − \frac{⟨\underline{a}_{i}, k̂ × \underline{â}_{j}⟩}{⟨\underline{x}_{i} − \underline{x}_{j}, k̂ × \underline{â}_{j}⟩}\underline{x}_{i})

Projection of the rolling compatibility condition onto the normal direction \underline{â}_{j} yields the rotation rate of the undriven wheel:

|\underline{a}_{j}| = ⟨\underline{a}_{i}, \underline{â}_{j}⟩ + Ω⟨\underline{x}_{i} − \underline{x}_{j}, \underline{â}_{j}⟩ = ⟨\underline{a}_{i}, \underline{â}_{j}⟩ − ⟨\underline{a}_{i}, k̂ × \underline{â}_{j}⟩\frac{⟨\underline{x}_{i} − \underline{x}_{j}, \underline{â}_{j}⟩}{⟨\underline{x}_{i} − \underline{x}_{j}, k̂ × \underline{â}_{j}⟩}

We have now solved the direct kinematics problem.

Example. Consider a tricycle WMR design, with one steered and driven front wheel, and two unsteered, undriven rear wheels placed symmetrically on a single axle, which is perpendicular to the straight line joining the front axle point with the center of mass of the WMR. We apply the preceding analysis to obtain the rotation rates |\underline{a}_{r1}|, |\underline{a}_{r2}| of the rear wheels, and the state trajectory (\underline{v}, Ω), in terms of the axle control \underline{a}_{f} for the front wheel. Let \underline{â}_{r} denote the common direction vector for the rear wheels, as shown.

INCLUDE F IGURE

We then have for k = 1, 2

|\underline{a}_{rk}| = ⟨\underline{a}_{f}, \underline{â}_{r}⟩ − ⟨\underline{a}_{f}, k̂ × \underline{â}_{r}⟩\frac{⟨\underline{x}_{f} − \underline{x}_{rk}, \underline{â}_{r}⟩}{⟨\underline{x}_{f} − \underline{x}_{rk}, k̂ × \underline{â}_{r}⟩}

Letting \underline{x}_{r} denote the midpoint of the rear axle, we have, by WMR design,

⟨\underline{x}_{f} − \underline{x}_{r1}, \underline{â}_{r}⟩ = \frac{1}{2}|\underline{x}_{r2} − \underline{x}_{r1}|, ⟨\underline{x}_{f} − \underline{x}_{r2}, \underline{â}_{r}⟩ = −\frac{1}{2}|\underline{x}_{r2} − \underline{x}_{r1}|

and also, for both k = 1, 2,

⟨\underline{x}_{f} − \underline{x}_{rk}, k̂ × \underline{â}_{r}⟩ = −|\underline{x}_{f} − \underline{x}_{r}|

Thus, the rotation rates become

|\underline{a}_{r1}| = ⟨\underline{a}_{f}, k̂ × \underline{â}_{r}⟩ + ⟨\underline{a}_{f}, k̂ × \underline{â}_{r}⟩ \frac{\frac{1}{2}|\underline{x}_{r2} − \underline{x}_{r1}|}{|\underline{x}_{f} − \underline{x}_{r}|}

|\underline{a}_{r2}| = ⟨\underline{a}_{f}, k̂ × \underline{â}_{r}⟩ − ⟨\underline{a}_{f}, k̂ × \underline{â}_{r}⟩ \frac{\frac{1}{2}|\underline{x}_{r2} − \underline{x}_{r1}|}{|\underline{x}_{f} − \underline{x}_{r}|}

Similarly, the state variables are found by direct kinematics to be

\underline{v} = −k̂ × (\underline{a}_{f} + \frac{⟨\underline{a}_{f}, k̂ × \underline{â}_{r}⟩}{|\underline{x}_{f} − \underline{x}_{r}|}\underline{x}_{f}) , Ω = \frac{⟨\underline{a}_{f}, k̂ × \underline{â}_{r}⟩}{|\underline{x}_{f} − \underline{x}_{r}|}.

1.2 Dynamics

Coulomb friction. The force on the i-th wheel is, according to Coulomb’s law of friction, F_{i} = −α_{i}u_{i}, where α_{i} is the net coefficient of friction and u_{i} is the virtual unit slippage velocity, which satisfies the constraint:

(F C_{i}) |u_{i}| ≤ 1

There are actually two Coulomb laws of friction. If the ith wheel is rolling at time t, then according to Coulomb’s static law, the contact force F_{i}(t) must satisfy the Euclidean norm constraint |F_{i}(t)| ≤ α_{i}, but is otherwise unconstrained. If the ith wheel is sliding at time t, then Coulomb’s dynamic law of friction states that the contact force F_{i}(t) is completely determined as a 2-vector by the state at time t and the

angular velocity a_{i}(t), and is given by F_{i}(t) = −α_{i} (v(t) + k̂ × (Ω(t)x_{i}(t) + a_{i}(t)))^{∧}.

By letting u_{i}(t) := (v(t) + k̂ × (Ω(t)x_{i}(t) + a_{i}(t)))^{∧} when the i-th wheel is slipping, and interpreting u_{i}(t) as an arbitrary vector of norm less than or equal to 1 when the ith wheel is rolling, we have F_{i}(t) = −α_{i}u_{i}(t) in both cases.

Dynamics. Taking into account the above expression for the frictional Coulomb force on the i-th wheel, we see that Newton’s equations of motion for the WMR, in the ground frame, are:

(Dyn)  \begin{aligned}v̇ & = −\frac{1}{m} ∑_{i} α_{i}u_{i} \\ Ω̇ & = −\frac{1}{I} ∑_{i} α_{i}⟨u_{i}, k̂ × R(θ)\underline{x}_{i}⟩ \\ ẋ & = v \\ θ̇ & = Ω \end{aligned}

We include the equations for ẋ, θ̇ explicitly because the boundary conditions for the minimum-time problem involve the values of x, θ. In the body frame, the dynamical equations become:

(\underline{Dyn})  \begin{aligned}˙\underline{v} & = −\frac{1}{m} ∑_{i} α_{i}\underline{u}_{i} − Ωk̂ × \underline{v} \\ Ω̇ & = −\frac{1}{I} ∑_{i} α_{i}⟨\underline{u}_{i}, k̂ × \underline{x}_{i}⟩ \\ ˙\underline{x} & = \underline{v} − Ωk̂ × \underline{x} \\ θ̇ & = Ω \end{aligned}

Dependence of slippage vectors \underline{u}_{i} (and thus of Coulomb frictional forces −α_{i}\underline{u}_{i}) on the angular velocities \underline{a}_{i}. Let us now ask to what extent the virtual slippage velocities \underline{u}_{i} are determined by the angular velocities \underline{a}_{i}. The force on the i-th wheel is in principle completely determined by the angular velocity functions a_{i}(·) only when the wheel is sliding. Simple quasi-static examples for bicycle WMR designs show that, in general, this indeterminacy cannot be resolved without invoking additional assumptions on the behavior of the frictional forces, beyond Coulomb’s law. In particular, the “equations” of motion are really differential inclusions if the \underline{a}_{i} are considered to be the controls (this is another motivation for our choice of the \underline{u}_{i} as the controls).

The dynamics of Coulomb friction involve less indeterminacy than the quasistatics. We will show that use of the WMR dynamics (\underline{Dyn}) in conjunction with the time derivative of the rolling constraint (\underline{RC}_{i}) (for the appropriate wheel type) leads to an affine constraint on the \underline{u}_{j} if the i-th wheel is rolling. Nonsingularity of the resulting nonhomogeneous linear system, contingent upon WMR design, allows one to obtain the \underline{u}_{i} from the \underline{a}_{i}.

The general rolling constraint for the i-th wheel is

(\underline{RC}_{i}(t)) : \underline{v}(t) + k̂ × (Ω(t)\underline{x}_{i}(t) + \underline{a}_{i}(t)) = 0

Substitution of the expressions for the time derivative of \underline{v} and Ω from (\underline{Dyn}) into this equation yields a dynamic rolling constraint on the i-th wheel:

(\underline{DRC}_{i}^{d})  \frac{1}{m} ∑_{j} α_{j}\underline{u}_{j} + \frac{1}{I} ∑_{j} α_{j}⟨\underline{u}_{j}, k̂ × \underline{x}_{j}⟩k̂ × \underline{x}_{i} = k̂ × ˙ \underline{a}_{i} − Ωk̂ × \underline{v}

This provides two scalar degrees of constraint on the \underline{u}_{j} for each rolling driven wheel. For undriven wheels, at most the direction \underline{â}_{i} may be specified by steering, not the magnitude |\underline{a}_{i}|. Thus, we must obtain a rolling constraint involving only the direction. We start from the kinematic rolling condition for an undriven wheel:

(\underline{RC}_{i}^{ud}) ⟨\underline{v} + Ωk̂ × \underline{x}_{i}, \underline{â}_{i}⟩ = 0

Differentiation and use of the dynamics, as before, yields the dynamic rolling constraint for undriven wheels:

(\underline{DRC}_{i}^{ud}) ∑_{j=1}^{N} α_{j}⟨\underline{u}_{j}, \frac{1}{m}\underline{â}_{i} + \frac{1}{I}⟨\underline{â}_{i}, k̂ × \underline{x}_{i}⟩k̂ × \underline{x}_{j}⟩ + Ω⟨k̂ × \underline{v}, \underline{â}_{i}⟩ = ⟨\underline{v} + Ωk̂ × \underline{x}_{i}, \underline{˙â}_{i}⟩

The time derivative of the direction, appearing on the right side of this equation, disappears if the wheel is unsteered. In any case, only one scalar degree of constraint on the \underline{u}_{j} is obtained from each rolling undriven wheel. Fortunately, this is the same as the degree of ambiguity in the force on the undriven wheels, as the direction of the force is always parallel to the axle for such wheels.

In summary, the rolling constraints give us, in the presence of the dynamics, a linear system for the forces \underline{u}_{1}, ..., \underline{u}_{r} of the rolling wheels in terms of the remaining \underline{u}_{i} (for the sliding wheels, i = r + 1, ..., N) and of the state (\underline{v}, Ω) and the angular controls \underline{a}_{i}, ˙ \underline{a}_{i}, \underline{â}_{i}, ˙ \underline{â}_{i} for the rolling wheels. Each undriven wheel contributes one scalar equation to this system, and each driven wheel contributes two scalar equations; thus, the dimension count is just right, since for an undriven wheel the direction of the force is determined by the corresponding (body-fixed) axle vector and there is one degree of freedom in choosing the scalar which gives the magnitude (modulo sign) of the force, while for a driven wheel the force has two degrees of freedom. If the system has maximum rank, then one may obtain the forces \underline{u}_{1}, ..., \underline{u}_{r} for the rolling wheels, in terms of the state of the WMR and of the angular velocities of the rolling wheels and their time derivatives. Recall that undriven wheels mounted on the same axle produce linearly dependent rolling constraints, so ambiguity in the forces will arise if there is more than one undriven wheel on any one axle. Any remaining ambiguity will be removed by the requirement of time-optimality, as we show in the next section.

Example.

We illustrate the preceding observations in the case of a bicycle WMR design. The front wheel is driven and steered, and the rear wheel, placed on the line joining the front axle point with the CM so that its axle is perpendicular to this line, is undriven and unsteered.

The dynamic rolling constraint associated with the front wheel is:

\frac{1}{m}(α_{f} \underline{u}_{f}+α_{r}|\underline{u}_{r}|\underline{â}_{r})+\frac{1}{I} (α_{f}⟨\underline{u}_{f}, k̂ × \underline{x}_{f}⟩k̂ × \underline{x}_{f} + α_{r}⟨|\underline{u}_{r}|\underline{â}_{r}, k̂ × \underline{x}_{r}⟩k̂ × \underline{x}_{f}) = k̂×˙\underline{a}_{f}−Ωk̂×\underline{v}

Since k̂ × \underline{x}_{f} = |\underline{x}_{f}|ˆ\underline{a}_{r}, and k̂ × \underline{x}_{r} = −|\underline{x}_{r}|ˆ\underline{a}_{r}, this becomes (\underline{RC}_{f}^{d})

\frac{1}{m}(α_{f} \underline{u}_{f} + α_{r}|\underline{u}_{r}|\underline{â}_{r}) + \frac{1}{I} (α_{f}|\underline{x}_{f}|^{2}⟨\underline{u}_{f}, ˆ \underline{a}_{r}⟩\underline{â}_{r} − α_{r}|\underline{u}_{r}||\underline{x}_{r}||\underline{x}_{f}|\underline{â}_{r}) = k̂ × ˙ \underline{a}_{f} − Ωk̂ × \underline{v}

Similarly, the rolling constraint for the rear wheel is:

(\underline{RC}_{r}^{ud})  α_{r}|\underline{u}_{r}|(\frac{1}{m} + \frac{1}{I}|\underline{x}_{r}|^{2}) + ⟨\underline{u}_{f}, \underline{â}_{r}⟩(\frac{1}{m} − \frac{1}{I}|\underline{x}_{r}||\underline{x}_{f}|) = −Ω⟨k̂ × \underline{v}, \underline{â}_{r}⟩

Choosing the oriented orthonormal frame \underline{â}_{r}, k̂ × \underline{â}_{r}, the matrix representing the homogeneous linear system associated with the above constraints, in the variables α_{r}|\underline{u}_{r}|, α_{f} \underline{u}_{f}, becomes

A = \frac{1}{mI} ( \begin{matrix}I − m|\underline{x}_{f}||\underline{x}_{r}| & I + m|\underline{x}_{f}|^{2} & 0 \\ 0 & 0 & \frac{1}{m} \\ I + m|\underline{x}_{r}|^{2} & I − m|\underline{x}_{f}||\underline{x}_{r}| & 0 \end{matrix} )

The determinant of this matrix is the square of (|\underline{x}_{f}| + |\underline{x}_{r}|). Thus, unless both wheels are placed at the center of mass, we may uniquely solve the nonhomogeneous system for \underline{u}_{f}, |\underline{u}_{r}| in terms of ˙ \underline{a}_{f} and of the state. Furthermore, by calculating the inverse matrix

A^{−1} = \frac{1}{(|\underline{x}_{f}| + |\underline{x}_{r}|)^{2}} ( \begin{matrix}−I + m|\underline{x}_{f}||\underline{x}_{r}| & 0 & −I + m|\underline{x}_{f}|^{2} \\ I + m|\underline{x}_{r}|^{2} & 0 & −I + m|\underline{x}_{f}||\underline{x}_{r}| \\ 0 & m(|\underline{x}_{f}| + |\underline{x}_{r}|)^{2} & 0 \end{matrix} )

and solving for the forces,

( \begin{matrix}α_{r}|\underline{u}_{r}| \\ α_{f} \underline{u}_{f} \end{matrix} ) = A^{−1} ( \begin{matrix}k̂ × ˙ \underline{a}_{f} − Ωk̂ × \underline{v} \\ −Ω⟨k̂ × \underline{v}, ˆ \underline{a}_{r}⟩ \end{matrix} ) ,

we obtain, from the Coulomb friction constraints −1 ≤ |\underline{u}_{r}| ≤ 1, |\underline{u}_{f}| ≤ 1, bounds on ˙ \underline{a}_{f}, Ω, \underline{v}, which must be satisfied if ideal rolling is to take place. Calculation yields the following explicit frictional bounds:

|⟨(−α + β_{f})Ω\underline{v} + α˙\underline{a}_{f}, k̂ × ˆ \underline{a}_{r}⟩| ≤ α_{r},
⟨β_{r} ˙\underline{a}_{f} + (β_{r} − α)Ω\underline{v}, k̂ × ˆ \underline{a}_{r}⟩^{2} + ⟨γ(˙\underline{a}_{f} − Ω\underline{v}), ˆ \underline{a}_{r}⟩^{2} ≤ α_{f},

where the constants α, β_{f}, β_{r}, γ are the nonzero entries of the inverse matrix A^{−1}: (see above)

A^{−1} = ( \begin{matrix}−α & 0 & β_{f} \\ β_{r} & 0 & −α \\ 0 & γ & 0 \end{matrix} )

One may also express the frictional bounds solely in terms of \underline{a}_{f}, ˙ \underline{a}_{f}, by substituting for the state variables \underline{v}, Ω, their expressions in terms of the axle vector \underline{a}_{f}, as given by direct kinematics. In fact, these expressions are precisely those obtained for the tricycle example in the section on kinematics. The resulting bounds are nonlinear differential inequalities for the axle vector function \underline{a}_{f}.

If instead of a bicycle WMR design we consider the tricycle design discussed before, we find that the rolling constraints for the rear wheels are linearly dependent, and thus that the frictional forces are not uniquely determined by the angular velocity of the front wheel and the dynamic state of the WMR. However, the sum of the forces on the rear wheels may be found by the preceding argument, and so certain bounds on the axle vector function for the front wheel may again be obtained.

2. Time–Optimal Control of WMRs

We now consider the problem of controlling a WMR in such a way as to minimize the total time required to transfer the state of the WMR between given initial and final states. We first examine, in section 2.1, the simpler case in which all of the wheels are fully controllable (steerable and drivable). We state the minimum–time problem and apply the Pontryagin maximum principle to find optimality conditions. Section 2.2 contains examples of closed–form solutions obtainable from the optimality conditions derived in section 2.1. Then in section 2.3 we go on to study the general case in which not all wheels are fully controllable.

2.1. Fully Controllable Wheels

In this section we formulate the minimum time problem for WMRs with fully controllable wheels and proceed to apply the Pontryagin maximum principle to find optimality conditions in the form of a two–point boundary value problem for the adjoint dynamics. In the present case of fully controllable wheels, the system of optimality conditions decouples into independent equations for each of the various wheels and the usual constrained optimization problem arising from the Pontryagin principle need not be explicitly considered. This simplification will no longer hold in the more general case in which not all wheels are fully controllable (see section 2.3).

2.1.1. Problem Statement.

Controls. We think of the vectors u_{i} as the “controls”, though it may actually be the angular velocity vectors a_{i} which may be adjusted through the driving mechanisms of the WMR. We assume u_{i}(t) to be a piecewise continuous function of t for each i. Depending on details of vehicle design, some of the u_{i} may be adjusted arbitrarily (driven, steerable wheels) while others may be constrained statically and/or dynamically to some degree (e.g., a wheel with fixed body orientation, or a “heavy” wheel with considerable steering inertia). The dynamically constrained case requires additional equations for the axle dynamics; this case will not be considered here. We will consider two types of wheels, both of which are assumed to be massless—one type is fully steerable and drivable (like the front wheel of a typical bicycle) and the other is undriven and unsteerable, i.e. its orientation in the WMR body is fixed (like a bicycle rear wheel). Each type of wheel gives rise to different control constraints. For every i, the definition of u_{i} requires that |u_{i}| ≤ 1; the possibility of strict inequality is included in order to cover the case in which the expression appearing in the numerator and denominator defining u_{i} vanishes. As discussed in [1], this occurs precisely when the i-th wheel is undergoing “ideal rolling;” the Coulomb frictional force then corresponds to a vector of arbitrary direction but bounded norm. If the i-th wheel is fully steerable and drivable, then this bound on the norm of u_{i} is the only control constraint. If, on the other hand, the i-th wheel is undriven, then since it is massless, no nonzero torque may be applied about its axle, and so the net frictional force must be perpendicular to the plane of the wheel, or, equivalently, parallel to the angular velocity vector a_{i}. In summary, the control constraints are:

(CC)

u_{i}(t) ≤ 1 for all t
u_{i}(t)||a_{i}(t) if the i-th wheel is not driven

It is convenient to rewrite the dynamics so that the terms involving driven wheels are separated from those corresponding to passive wheels. We will use the subscript i to refer to driven wheels and j for fixed wheels; â_{j} denotes the unit vector in the direction of the angular velocity vector a_{j}:

(Dyn)  \begin{aligned}v̇ & = \frac{1}{m} ∑_{i} α_{i}u_{i} + \frac{1}{m} ∑_{j} α_{j}â_{j} \\ Ω̇ & = \frac{1}{I} ∑_{i} α_{i}⟨u_{i}, k̂ × R(θ)\underline{x}_{i}⟩ + \frac{1}{I} ∑_{j} α_{j}⟨â_{j}, k̂ × R(θ)\underline{x}_{j}⟩ \\ ẋ & = v \\ θ̇ & = Ω \end{aligned}

In body coordinates, we obtain:

(\underline{Dyn})  \begin{aligned}˙\underline{v} & = −Ωk̂ × \underline{v} + \frac{1}{m} ∑_{i} α_{i}\underline{u}_{i} + \frac{1}{m} ∑_{j} α_{j}\underline{â}_{j} \\ Ω̇ & = \frac{1}{I} ∑_{i} α_{i}⟨\underline{u}_{i}, k̂ × \underline{x}_{i}⟩ + \frac{1}{I} ∑_{j} α_{j}⟨\underline{â}_{j}, k̂ × \underline{x}_{j}⟩ \\ ˙\underline{x} & = −Ωk̂ × \underline{x} + \underline{v} \\ θ̇ & = Ω \end{aligned}

Boundary Conditions. We assume boundary conditions, in ground coordinates, of the form

(BC) [ \begin{matrix}v(0) \\ Ω(0) \\ x(0) \\ θ(0) \end{matrix} ] = [ \begin{matrix}v_{0} \\ Ω_{0} \\ x_{0} \\ θ_{0} \end{matrix} ], [ \begin{matrix}v(t_{f}) \\ Ω(t_{f}) \\ x(t_{f}) \\ θ(t_{f}) \end{matrix} ] = [ \begin{matrix}v_{f} \\ Ω_{f} \\ x_{f} \\ θ_{f} \end{matrix} ]

(fixed endpoints)

The initial “time” is zero, and the final time t_{f} is unknown. In fact, the problem we are interested in is precisely the

Time-Optimal Problem. Find piecewise continuous controls u_{i}(·) satisfying the control constraints (CC) so that the system with dynamics (Dyn) meets the boundary conditions (BC), in such a way that the final time t_{f} is minimized.

2.1.2. Optimality Conditions

A standard form of Pontryagin’s maximum principle (see [6]) yields necessary conditions for optimality of controls u_{1}, …, u_{W} (W = total number of wheels) for the minimum time problem stated at the end of the preceding section.

Specifically, consider the pseudo-Hamiltonian function

\begin{aligned}(H) : H & ([ \begin{matrix}v \\ Ω \\ x \\ θ \end{matrix} ] , [ \begin{matrix}v^{∗} \\ Ω^{∗} \\ x^{∗} \\ θ^{∗} \end{matrix} ] , [ u_{1} ; u_{2} ; ⋮ ; u_{W} ]) = ⟨[ \begin{matrix}v^{∗} \\ Ω^{∗} \\ x^{∗} \\ θ^{∗} \end{matrix} ] , [ \begin{matrix}\frac{1}{m} ∑_{i} α_{i}u_{i} \\ \frac{1}{I} ∑_{i} α_{i}⟨u_{i}, k̂ × R(θ)\underline{x}_{i}⟩ \\ v \\ Ω \end{matrix} ]⟩ \\ & = ∑_{i} α_{i}⟨u_{i}, \frac{1}{m}v^{∗} + \frac{1}{I}Ω^{∗}k̂ × R(θ)\underline{x}_{i}⟩ + ⟨x^{∗}, v⟩ + θ^{∗}Ω, \end{aligned}

where ⟨·, ·⟩ is the standard inner product in ℝ^{2}.

Consider also the adjoint system

[ v̇^{∗} ; Ω̇^{∗} ; ẋ^{∗} ; θ̇^{∗} ] = −\frac{∂(Dyn)^{T}}{∂(v, Ω, x, θ)} [ \begin{matrix}v^{∗} \\ Ω^{∗} \\ x^{∗} \\ θ^{∗} \end{matrix} ]

where Dyn denotes the function corresponding to the dynamics (Dyn) of the previous section, i.e.,

Dyn = [ \begin{matrix}\frac{1}{m} ∑_{i} α_{i}u_{i} \\ \frac{1}{I} ∑_{i} α_{i}⟨u_{i}, k̂ × R(θ)\underline{x}_{i}⟩ \\ v \\ Ω \end{matrix} ]

(this is the same function which appears in the definition of the pseudo-Hamiltonian above).

We obtain the adjoint dynamics:

(Dyn^{∗})  \begin{aligned}v̇^{∗} & = −x^{∗} \\ Ω̇^{∗} & = −θ^{∗} \\ ẋ^{∗} & = 0 \\ θ̇^{∗} & = \frac{1}{I} ∑_{i} α_{i}⟨u_{i}, R(θ)\underline{x}_{i}⟩⋮Ω^{∗} \end{aligned}

The maximum principle states that a necessary condition for optimality of a control ũ = (ũ_{1}, …, ũ_{W}) in the class of piecewise continuous functions [0, t_{f}] → ℝ^{W}, is the existence of a nonzero solution (v^{∗}, Ω^{∗}, x^{∗}, θ^{∗}) of the adjoint dynamics (Dyn^{∗}) such that for all times t ∈ [0, t_{f}] except possibly the points of discontinuity of ũ(·), the instantaneous value ũ(t) of the control solves the following finite-dimensional optimization problem for the Hamiltonian (H), the trajectory associated to ũ(·) by the dynamics (Dyn) and the boundary conditions (BC):

max_{\begin{matrix}|u_{i}|≤1 \\ i=1,...,W \end{matrix}} H ([ \begin{matrix}v(t) \\ Ω(t) \\ x(t) \\ θ(t) \end{matrix} ] , [ \begin{matrix}v^{∗}(t) \\ Ω^{∗}(t) \\ x^{∗}(t) \\ θ^{∗}(t) \end{matrix} ] , [ \begin{matrix}u_{1} \\ u_{2} \\ . & . & . \\ u_{W} \end{matrix} ] , )

Using the explicit expression (H) available for the pseudo-Hamiltonian H, we see that this is equivalent to maximizing the sum

∑_{i=1}^{W} α_{i}⟨u_{i}, \frac{i}{m}v^{∗}(t) + \frac{1}{I}Ω^{∗}(t)k̂ × R(θ(t))\underline{x}_{i}⟩

subject to |u_{i}| ≤ 1.

Fortunately, we may maximize each summand independently of the others. The solution is:

(OC)  ũ_{i}(t) = \frac{\frac{1}{m}v^{∗}(t) + \frac{1}{I}Ω^{∗}(t)k̂ × R(θ(t))\underline{x}_{i}}{|\frac{1}{m}v^{∗}(t) + \frac{1}{I}Ω^{∗}(t)k̂ × R(θ(t))\underline{x}_{i}|}

In summary, the maximum principle tells us that the optimal controls should be of the above from (OC), where the state (v, Ω, x, θ) and the adjoint state (v^{∗}, Ω^{∗}, x^{∗}, θ^{∗}) satisfy the coupled systems (Dyn) + (BC) and (Dyn^{∗}) which are driven by the ũ_{i}(·).

After removing the essentially superfluous equations corresponding to x^{∗} and v^{∗} in (Dyn^{∗}), we arrive at the enlarged system of dynamics and adjoint dynamics:

(ODyn)

v̇ = \frac{1}{m} ∑_{i} α_{i} \frac{\frac{1}{m}v^{∗}(t) + \frac{1}{I}Ω^{∗}(t)k̂ × R(θ(t))\underline{x}_{i}}{|\frac{1}{m}v^{∗}(t) + \frac{1}{I}Ω^{∗}(t)k̂ × R(θ(t))\underline{x}_{i}|}

Ω̇ = \frac{1}{I} ∑_{i} α_{i}⟨\frac{\frac{1}{m}v^{∗}(t) + \frac{1}{I}Ω^{∗}(t)k̂ × R(θ(t))\underline{x}_{i}}{|\frac{1}{m}v^{∗}(t) + \frac{1}{I}Ω^{∗}(t)k̂ × R(θ(t))\underline{x}_{i}|}, k̂ × R(θ)\underline{x}_{i}⟩
ẋ = v
θ̇ = Ω
Ω̇^{∗} = −θ^{∗}
θ̇^{∗} = \frac{1}{I} ∑_{i} α_{i}⟨\frac{\frac{1}{m}v^{∗}(t) + \frac{1}{I}Ω^{∗}(t)k̂ × R(θ(t))\underline{x}_{i}}{|\frac{1}{m}v^{∗}(t) + \frac{1}{I}Ω^{∗}(t)k̂ × R(θ(t))\underline{x}_{i}|}, R(θ)\underline{x}_{i}⟩ · Ω^{∗}

where v^{∗}(t) = a + tb for arbitrary parameters a, b ∈ ℝ^{2}, and it is assumed that the boundary conditions (BC) are satisfied. We have six degrees of freedom in choosing a, b, Ω^{∗}(0), θ^{∗}(0) to drive the system from the given initial conditions to the 6-parameter final conditions.

2.2 Analytical Results

In this section we use the differential system (ODyn) found above, to investigate the optimal trajectories for specific WMR designs having only fully controllable wheels. General closed form solutions are given for a robot with a single wheel at the center of mass. For a robot with a single wheel, but not at the center of mass, special solutions are found in closed form, exhibiting interesting behavior. These special solutions include certain of the spirals which Kanayama and Hartman (1989) found as extremals for a cost function involving only geometric properties of the associated path in position-space. We show that the members of a subclass of the class of these spirals, traversed at constant speed, are in fact extremals for the minimum time problem, for which, in contrast to the case of the previously mentioned cost function, the dynamics of the WMR do play a role. The optimal controls for a more general WMR with fully controllable wheels may be decomposed into a ground term which is constant in the ground frame and a body term which is constant in the body frame. Special solutions may be found by assuming that only one of these terms is nonzero. The behavior of the solutions when only the ground term is present is similar to that for a pointmass WMR, while presence of only the body term leads to solutions similar to the previously described spirals. We hope that the decomposition into ground and body terms will allow a qualitative description of the general class of optimal trajectories for WMRs with fully controllable wheels, but at present we do not know how to do this.

2.2.1. Spacetime transformations yielding new solutions from old.

Observe that in the ordinary differential system (ODyn) from the preceding section, time t and the constant 2-vector b appear only in the combination tb.

This leads to special one-parameter families of optimal trajectories, as follows. Let a(c, b, t) denote the acceleration v̇(t) along the trajectory corresponding to the given constant vectors c, b, and let x(c, b, t) denote the associated position, satisfying

ẍ(c, b, t) = a(c, b, t), ẋ(c, b, 0) = v_{0}, x(c, b, 0) = x_{0}

where v_{0}, x_{0} are, respectively, the fixed initial velocity and position.

Then a(c, μb, t) = a(c, b, μt) for any real μ, and by repeated integration we find:

x(c, μb, t) = μ^{−2}x(c, b, μt) + (1 − μ^{−1})tv_{0} + (1 − μ^{−2})x_{0}

Thus, the effect of scaling b is to effect a Galilean transformation of space-time following a scaling of space-time which includes a possible time reversal. In other words, the trajectory corresponding to the scaled b is a scaled version of the trajectory for the original b, viewed in a frame which is moving at a constant velocity with respect to the original ground frame, and possibly traversed in the opposite direction.

Another such, even more obvious symmetry of (ODyn), is its invariance under orientation-preserving homotheties of the adjoint state space {(v^{∗}, Ω^{∗}, x^{∗}, θ^{∗})}, i.e. under simultaneous scaling of all adjoint variables by the same positive real factor. In verifying this property, recall that the constants c, b are given by b = x^{∗}(0), c = v̇^{∗}(0). Since the initial value of the state (v, Ω, x, θ) is fixed, the initial value of the costate determines the trajectory via the equation of evolution (ODyn). Invariance under positive homotheties of the adjoint state therefore suggests that we think of the family of optimal solutions as being parametrized not by points in ℝ^{6}, but rather by points on the 5-sphere S^{5} = {x ∈ ℝ^{6} : |x| = 1}. This fact was used in implementing a shooting method for numerically solving the two point boundary value problem for (ODyn) in the case of a tricycle, as described in the preceding section.

Finally, we observe that any rigid motion of the space ℝ^{2} in which the vector variables take their values, of course takes solutions to solutions.

2.2.2. A one-wheeled robot.

We consider the simple example of a WMR with one driven and steered wheel and two additional supporting castors which do not affect the dynamics.

Wheel at CM. First assume that the driven wheel is placed at the center of mass of the robot. Then the vector \underline{x}_{1} which joins the CM with the axle point of the wheel vanishes, so the body term disappears from (ODyn), leaving only

(ODyn_{pm})  \begin{aligned}v̇ & = −\frac{α}{m}(c + tb)^{∧} \\ ẋ & = v \end{aligned}

The angular variables Ω, θ are irrelevant, since no torque can be exerted on the WMR about the vertical through the CM for this design. We may think of this WMR with one wheel at the CM as a point-mass WMR. The parameters c, b ∈ ℝ^{2} must be chosen so that the final condition (v(t_{f}), x(t_{f})) = (v_{f}, x_{f}) is satisfied, assuming that the initial state is (v(0), x(0)) = (v_{0}, x_{0}), as in (BC).

The minimum-time problem for a point-mass WMR is similar to the customary example given in textbooks on optimal control (see e.g. Leitmann (1981) Cesari(19– )) except that the custom is to assume that the components u_{1}, u_{2} of the control satisfy decoupled norm constraints, leading to optimal controls of bang-bang type.

The discussion preceding this example shows that we may assume that c and b both have unit length (i.e. we think of ℝ^{4} as the product of two copies of ℝ^{2}, and we consider the unit sphere in the norm ||(c, b)|| = max {c, b}). Furthermore, it is clear that any rotation of ℝ^{2} acting simultaneously on all of the vector variables, from both the state (including the initial conditions) and the costate, yields a new solution of the dynamics (ODyn_{pm}). Thus, modulo such a rotation, we may assume that the vector c coincides with a given fixed unit vector, say the canonical basis vector î in ℝ^{2}, and so we see that all trajectories are determined, modulo a rotation, by an arbitrary choice of b on the unit circle S^{1}. However, even this restricted class of trajectories is redundant: there are in fact only two essentially different kinds of trajectories for (ODyn_{pm}), corresponding to the cases in which c, b are, respectively, parallel or perpendicular, as the following argument shows. Suppose that c, b are nonparallel. Then in the usual way we may find mutually perpendicular vectors c′, b′ with the same linear span as c, b, by letting c′ = c − \frac{⟨c,b⟩}{⟨b,b⟩}b, b′ = b, and we

see that if t′ = t + \frac{⟨c,b⟩}{⟨b,b⟩}, then a(c, b, t) = a(c′, b′, t′), so the net result of assuming that c, b are in fact perpendicular is that we must allow for a time translation; in particular, negative times must be considered. We now describe the two kinds of trajectories for a pointmass WMR. Case 1. c = b.

Then the acceleration is \{ \begin{matrix}\frac{α}{m}c & if t < −1 \\ −\frac{α}{m}c & if t > −1 \end{matrix}\}

Thus, in this case we get constant acceleration, with the option of switching sign once at some time (if the “original” c, b are antiparallel). In particular, the path traced out in position-space in this case is the union of two parabolas whose axes of symmetry are parallel. It may be shown that the set of final states which may be reached from a given initial state by choosing parallel c, b as in Case 1 has zero Lebesgue measure in ℝ^{4}; thus, most of the time we will be in the following case. Case 2. c ⊥ b, |c| = 1 = |b|.

The acceleration is − \frac{α}{m} · \frac{c + tb}{|c + tb|} = −\frac{α}{m} · \frac{c + tb}{\sqrt{1 + t^{2}}} = −\frac{α}{m}(cos β · c + sin β · b)

where β = arctan t is the polar angle in the (c, b)-coordinate plane, i.e. the acceleration corresponds to a rotating vector of constant length \frac{α}{m}, which tracks a point moving with unit speed along a straight line parallel to b and lying at unit distance from the origin of the (c, b) plane. By retracing the steps which led us to the present simplified cases 1 and 2 for c, b, we see that as we decrease the angle between the original c, b toward zero without letting c ever actually equal b, the corresponding acceleration vectors track points of higher and higher speed along the same line described above (the speed varies like |c × b|^{−1}). Thus, Case 1, in which c and b are parallel, may be seen as a limiting Case 2 in which the speed of the tracked point is infinite.

The above expression for the acceleration may be explicitly integrated to get closed-form expressions for both the velocity and the position as a function of time:

v(t) = v_{0} + log |t + \sqrt{1 + t^{2}}|\frac{α}{m}c + (\sqrt{1 + t^{2}} − 1)\frac{α}{m}b
x(t) = x_{0} + tv_{0} + (t log |t + \sqrt{1 + t^{2}}| − \sqrt{1 + t^{2}} + 1) \frac{α}{m}c

+ [\frac{1}{2} (t\sqrt{1 + t^{2}} + log |t + \sqrt{1 + t^{2}}|) − t] \frac{α}{m}b

If we choose c, b to be, respectively, horizontal and vertical unit vectors, let x_{0} be the origin, and fix the value of α/m to be 1, say, then the trajectory x(·) is determined by the initial velocity vector v_{0}. Typical trajectories are illustrated below.

INCLUDE MATHEMATICA GRAPHICS

The trajectories corresponding to arbitrary nonparallel c, b may be obtained from those for perpendicular unit c, b by applying the transformations described above. Specifically, any such trajectory may be obtained by starting with a trajectory associated with perpendicular pair of unit 2-vectors c, b and then applying a time translation (to compensate for non-perpendicularity of c, b) and a Galilean transformation composed with a time-space scaling (to compensate for deviation of the quotient |c|/|b| from unity). To be completely explicit, one has

x(c, b, t) = x(c′, b, t′)

where the new parameters are given by

c′ = c − \frac{⟨c, b⟩}{⟨b, b⟩}b, t′ = t + \frac{⟨c, b⟩}{⟨b, b⟩}

Invariance under positive scalings now yields

x(c, b, t) = x(ĉ′, \frac{|b|}{|c′|}b̂, t′)

Here, notice that

|c′| = \sqrt{|c|^{2} − \frac{⟨c, b⟩^{2}}{|b|^{2}}}

Thus, by letting

μ = (|c|^{2} − \frac{⟨c, b⟩^{2}}{|b|^{2}} )^{−1/2} ,

we conclude, taking into account the comments in paragraph 2.2.1, that

x(c, b, t) = μ^{−2}x (c′, b̂, μt′) + (1 − μ^{−1}) t′v_{0} + (1 − μ^{2}) x_{0}

Since c, b will generically be nonparallel, the above shows that, in the pointmass case, generic time-optimal trajectories may be obtained from knowledge of the family of trajectories corresponding to a single choice of nonparallel c, b, for all initial conditions on the state, by time translations, spacetime scalings, and Galilean transformations of spacetime, together with rotations of ℝ^{2} affecting all 2-vector variables simultaneously.

Wheel not at CM. Now consider the one-wheeled robot with the driven wheel not at the CM. In this case the frictional force on the wheel, in contrast to the case in which the wheel is at the CM, can exert a nonzero torque about the vertical through the CM. The system (ODyn) of optimal dynamics is

\begin{aligned}v̇ & = −\frac{α}{m} (\frac{1}{m}(c + tb) + \frac{1}{I}Ω^{∗}k̂ × R(θ)\underline{x})^{∧} \\ Ω̇ & = −\frac{α}{I} ⟨(\frac{1}{m}(c + tb) + \frac{1}{I}Ω^{∗}k̂ × R(θ)\underline{x})^{∧} , k̂ × R(θ)\underline{x}⟩ \\ ẋ & = v \\ θ̇ & = Ω \\ Ω̇^{∗} & = −θ^{∗} \\ θ̇^{∗} & = −\frac{α}{I} ⟨(\frac{1}{m}(c + tb) + \frac{1}{I}Ω^{∗}k̂ × R(θ)\underline{x})^{∧} , R(θ)\underline{x}⟩ · Ω^{∗} \end{aligned}

Recall that here the constants b, c are given by: b = x^{∗},c = v̇^{∗}. The constant vector \underline{x} is as shown in the drawing. The expression R(θ)\underline{x} in the equations represents the coordinates of \underline{x} as seen from the stationary ground frame.

The control term

u = (\frac{1}{m}(c + tb) + \frac{1}{I}Ω^{∗}k̂ × R(θ)\underline{x})^{∧}

which appears in the equations for v̇, Ω̇, and θ̇^{∗} consists of two parts. The first, \frac{1}{m}(c + tb) is the same as in the point-mass case. Its value is constant in the ground frame; we call this term the ground term. The second, \frac{1}{I}Ω^{∗}k̂ × R(θ)\underline{x}, is constant in the body frame, and is precisely perpendicular to the vector \underline{x} which joins the center of mass of the WMR with the axle point of the driven wheel. We refer to it as the body term.

Trajectories for the case in which the body term vanishes are similar to those for a point mass, except that now the angular motion must also be taken into account. More specifically, if only the ground term is present, then the center of mass of the present WMR moves as would a point mass WMR undergoing general time-optimal motion, while the angular motion is determined by the associated torques. The scaling arguments from the pointmass case, yielding generic optimal trajectories in terms of a single such, remain valid here. We will not study this case further.

Consider now the case in which the ground term vanishes, leaving only the body term

u = sgn(Ω^{∗})(k̂ × R(θ)\underline{x})^{∧}

The acceleration is

v̇ = −\frac{α}{m} · sgn(Ω^{∗})(k̂ × R(θ)\underline{x})^{∧}

The force on the i-th wheel thus has unit magnitude and is perpendicular to the vector joining the CM of the W MR with the i-th axle point at all times, with possible sign switches modulated by Ω^{∗}(t).

There is no reason to expect the velocity v to be parallel to the axle point vector R(θ)\underline{x} in general. In fact, the unicycle design we are discussing here is somewhat difficult to orient in this sense. However, some optimal trajectories are traversed in their entirety with v parallel to R(θ)\underline{x}, i.e., with the front of the robot body actually pointing forward all of the time. Still assuming that only the body term is present, let’s find such a trajectory. The magnitude |v| of the velocity must remain constant, because the acceleration will be perpendicular to v. The dynamics for the angle θ are:

\begin{aligned}θ̇ & = Ω \\ Ω̇ & = −\frac{α}{m}sgn(Ω^{∗}) · |\underline{x}|^{2} \end{aligned}

From the adjoint dynamics, we see that Ω^{∗} is linear in t and therefore changes sign at most once. On each of the two semi-infinite time intervals during which the sign of Ω^{∗} remains constant, we see that Ω̇ is also constant, so that the angle θ is of the form

(∗)  θ(t) = θ_{0} + Ω_{0}t + νt^{2}

for certain parameters θ_{0}, Ω_{0}, ν ∈ ℝ. The trajectory is being traversed at constant speed, and so we may identify time t with arclength s, modulo a constant scaling factor. Thus, the position-space path is a curve x(s) parameterized by arclength, such that the angle θ(s) between the tangent vector ẋ(s) and a fixed ground direction is quadratic, as in (∗) above. This is a quadratic spiral which belongs to the class of cubic spirals found by Kanayama and Hartman [2] as minimizers for the cost

∫_{0}^{s_{f}} (\frac{d^{2}θ}{ds^{2}} )^{2} ds

which is intended to measure the “nonsmoothness” of the path. This cost function depends solely on geometric properties of the path in position-space. Our analysis reveals the quadratic spirals to be locally time-minimizing trajectories for the unicycle WMR under special conditions, taking into account the relevant rigid-body dynamics and the Coulomb law of friction.

We do not know how to combine the above descriptions of the body term only and ground term only cases to obtain a complete qualitative understanding of the general time-optimal trajectories for a WMR with fully controllable wheels. Certain coarse statements may be made. For example, it is clear that only the body terms remain in the limit as the moment of inertia I of the WMR becomes infinite.

Furthermore, the solutions of the differential system (ODyn) should depend continuously on the parameter I. Thus, as I increases, one expects to observe that the family of optimal trajectories approaches a family in which all trajectories may be obtained from any single family member by rotations, time translations, spacetime scalings, and Galilean transformations.

2.3. Constrained Wheels

In this section, we address the minimum time problem for WMR designs with constrained, i.e. unsteered or undriven, wheels. For such wheels, the slippage vectors \underline{u}_{i} are no longer arbitrary subunit vectors, in contrast to the case for fully controllable wheels. We will explicitly find appropriate state-dependent constraints on the \underline{u}_{i}, associated with the constrained rolling wheels, which guarantee that the candidate \underline{u}_{i} are physically realizable as Coulomb frictional forces for the given wheel types. For such \underline{u}_{i}, there exist unique wheel angular velocities \underline{a}_{i}, namely those determined by inverse kinematics, giving rise to the same state trajectory as the original \underline{u}_{i}. Thus, the \underline{u}_{i} may still, as in the case of fully controllable wheels, be considered to be the controls. Application of an appropriate version of the Pontryagin maximum principle then yields a two point boundary value problem for the adjoint dynamics, involving a nested finite-dimensional optimization problem for an affine objective function, the pseudo-Hamiltonian, subject to convex constraints. Explicit solution of the finite-dimensional optimization problem, which was trivial in the case of fully controllable wheels (see section 2.1), is possible also for simple designs involving constrained wheels, though the resulting expressions are complicated. We illustrate a more generally applicable qualitative analysis of the optimization, in the case of a bicycle WMR with one fully controllable wheel and one constrained wheel.

2.3.1. Controls. As before, we regard the dynamics (Dyn) (repectively, (\underline{Dyn})) as an ordinary differential system in the dynamic state (v, Ω, x, θ) (respectively, (\underline{v}, Ω, \underline{x}, θ)) of the WMR, with the virtual slippage velocities u_{i} as the controls, subject to the (Coulomb friction) control constraint

(F C_{i}) |u_{i}| ≤ 1

(respectively, |\underline{u}_{i}| ≤ 1).

Wheels which are fully controllable in the sense that the associated a_{i} may be chosen to be any piecewise continuous function of time, give rise to no control constraints beside (F C_{i}). All other wheel types give rise to additional, generally state-dependent, control constraints. In deriving these control constraints, we will proceed much as in the discussion of the dependence of the slippage velocities \underline{u}_{i} on the angular velocities \underline{a}_{i}, in the section on dynamics. Namely, the kinematic rolling constraint for the i-th wheel, which states that the slippage velocity of the i-th wheel vanishes identically, is differentiated with respect to time, yielding a constraint involving time derivatives of the state variables; the dynamical equations of the WMR are then used to express these derivatives in terms of the state and of the \underline{u}_{i}, thus yielding the desired control constraint.

However, the objective here is to find constraints on the \underline{u}_{i} which guarantee that associated \underline{a}_{i} exist. The \underline{a}_{i} are now, in contrast to the discussion in the preceding section, considered to be unknown (they will actually turn out to be uniquely determined by inverse kinematics) and, accordingly, should not appear in the final constraints, with the exception that if the i-th wheel is unsteered, then the fixed direction vector \underline{â}_{i} is known and may appear in the corresponding constraint.

Dynamic rolling constraint for unsteered wheels. If the i-th wheel is driven but not steered, then only |\underline{a}_{i}| may be adjusted, not \underline{â}_{i}. Thus, \underline{a}_{i} may be adjusted to satisfy the rolling constraint \underline{v} + Ωk̂ × \underline{x}_{i}, = −k̂ × \underline{a}_{i}, iff

(\underline{RC}_{i}^{us}) ⟨\underline{v} + Ωk̂ × \underline{x}_{i}, \underline{â}_{i}⟩ = 0

The same may be said if the i-th wheel is unsteered and undriven. In that case, the adjustment of |\underline{a}_{i}| is made automatically by the massless wheel, so as to cancel the torque about its axle due to Coulomb friction.

Differentiation of the kinematic constraint (RC_{i}^{us}) with respect to time, and substitution of the dynamics (\underline{Dyn}), yields the equation:

(\underline{DRC}_{i}^{us}) 0 = ⟨˙\underline{v}+Ω̇k̂×\underline{x}_{i}, \underline{â}_{i}⟩ = ∑_{j=1}^{N} α_{j}⟨\underline{u}_{j}, \frac{1}{m}\underline{â}_{i}+ \frac{1}{I}⟨\underline{â}_{i}, k̂×\underline{x}_{i}⟩k̂×\underline{x}_{j}⟩+Ω⟨k̂×\underline{v}, \underline{â}_{i}⟩

which is an affine state-dependent constraint on \underline{u}_{1}, ..., \underline{u}_{N}.

Zero torque constraint for undriven wheels. If the i-th wheel is undriven, then it is required that the torque about the i-th axle vanish, i.e.

(\underline{T C}_{i}^{us,ud}) ⟨\underline{u}_{i}, k̂ × \underline{â}_{i}⟩ = 0

This is true whether the wheel is steered or not. However, if the wheel is steered , then \underline{â}_{i} is unknown, so we must eliminate \underline{â}_{i} from the torque constraint on the i-th wheel. We do this by using the rolling constraint (\underline{RC}_{i}), which states that \underline{v} + Ωk̂ × \underline{x}_{i} must be parallel to k̂ × \underline{â}_{i}. Substitution into the torque constraint (\underline{T C}_{i}^{ud}) yields:

(\underline{T C}_{i}^{s,ud}) ⟨\underline{u}_{i}, \underline{v} + Ωk̂ × \underline{x}_{i}⟩ = 0

Existence of wheel velocities \underline{a}_{i} corresponding to given slippage vectors \underline{u}_{i}. We have now stated the necessary control constraints on the \underline{u}_{i}, associated with different wheel types. We summarize as follows.

Proposition. Suppose \underline{u}_{1}, ..., \underline{u}_{N} are found as functions \underline{u}_{j} : [0, t_{f}] → ℝ^{2} satisfying the following conditions, where (\underline{v}, Ω, \underline{x}, θ) is the state trajectory generated by the \underline{u}_{j} via (\underline{Dyn}), and each condition is assumed to hold only for wheels of the appropriate type as explained above:

(\underline{F C}_{i})  |u_{i}| ≤ 1

(\underline{DRC}_{i}^{us})  ∑_{j=1}^{N} α_{j}⟨\underline{u}_{j}, \frac{1}{m}\underline{â}_{i} + \frac{1}{I}⟨\underline{â}_{i}, k̂ × \underline{x}_{i}⟩k̂ × \underline{x}_{j}⟩ + Ω⟨k̂ × \underline{v}, \underline{â}_{i}⟩ = 0

(\underline{T C}_{i}^{s,ud})  ⟨\underline{u}_{i}, \underline{v} + Ωk̂ × \underline{x}_{i}⟩ = 0

(\underline{T C}_{i}^{us,ud})  ⟨\underline{u}_{i}, k̂ × \underline{â}_{i}⟩ = 0

Assume furthermore that at the initial time t = 0, each rolling constraint is satisfied, i.e.

(\underline{RC}_{i}(0))  \underline{v}(0) + k̂ × (Ω(0)\underline{x}_{i}(0) + \underline{a}_{i}(0)) = 0

Then there exist unique wheel angular velocity functions

\underline{a}_{i} : [0, t_{f}] → ℝ^{2}, i = 1, ..., N

namely, those determined by inverse kinematics (IK) from the given \underline{u}_{i}, such that the state trajectory corresponding to these \underline{a}_{i} by direct kinematics (DK), is precisely the trajectory (\underline{v}, Ω, \underline{x}, θ) generated by the given \underline{u}_{i} via the dynamics (\underline{Dyn}). All wheels roll along this state trajectory.

Proof. Clearly, the \underline{a}_{i} determined by inverse kinematics are the only candidates. It suffices to show that these \underline{a}_{i} are consistent with the state trajectory associated with the given \underline{u}_{i}, i.e. that the rolling conditions (\underline{RC}_{i}) are satisfied for these \underline{a}_{j}, \underline{u}_{j}. The i-th rolling condition (\underline{RC}_{i}(t)) holds for all t, iff (\underline{RC}_{i}(0)) holds and also the time-derivative \frac{d}{dt}(\underline{RC}(t)) holds for all t. But this is guaranteed by the hypotheses of the Proposition, since the dynamic rolling constraint (\underline{DRC}_{i}^{us}) is, in the presence of the WMR dynamics (\underline{Dyn}), precisely equivalent to the time-derivative of the ith rolling constraint applied to the \underline{a}_{j} given by inverse kinematics. The remaining hypotheses ensure that the \underline{u}_{i} are consistent with individual wheel design.

Minimum-Time Problem. Taking into account the preceding discussion, we may now formulate the mimimum time problem for WMRs as follows. We wish to minimize t_{f} > 0 subject to the existence of piecewise continuous controls \underline{u}_{i} : [0, t_{f}] ⟶ ℝ^{2}, i = 1, ..., N satisfying, for each rolling wheel, the corresponding control constraint in the statement of the above Proposition, and which drive the system with dynamics

(\underline{Dyn})  \begin{aligned}˙\underline{v} & = −\frac{1}{m} ∑_{i} α_{i}\underline{u}_{i} − Ωk̂ × \underline{v} \\ Ω̇ & = −\frac{1}{I} ∑_{i} α_{i}⟨\underline{u}_{i}, k̂ × \underline{x}_{i}⟩ \\ ˙\underline{x} & = v − Ωk̂ × \underline{x} \\ θ̇ & = Ω \end{aligned}

from the state (\underline{v}_{0}, Ω_{0}, \underline{x}_{0}, θ_{0}) at time t = 0, to the state (\underline{v}_{f}, Ω_{f}, \underline{x}_{f}, θ_{f}) at time t = t_{f}.

2.3.2. Optimality Conditions.

We will apply the Pontryagin maximum principle, stated as follows (from Leitmann (1981), section 13.10).

Theorem. A necessary condition for time-optimality of the control u : [0, t_{f}] → ℝ^{m}, in the class of piecewise continuous functions, for the system with dynamics ẋ(t) = f(x(t), u(t), t) and fixed initial and final states x(0), x(t_{f}) ∈ ℝ^{n}, in the presence of the control constraint u(t) ∈ Γ, t ∈ [0, t_{f}], and of the additional statedependent control constraints ϕ_{i}(x(t), u(t)) ≤ 0, t ∈ [0, t_{f}], i = 1, ..., r, is the existence of a nonzero trajectory x^{∗} : [0, t_{f}] → ℝ^{n} of the adjoint dynamics

ẋ^{∗}(t) = −D_{x}f(x(t), u(t), t)^{T} x^{∗}(t) + (D_{x}Φ^{T})(D_{u_{1},..,u_{r}}Φ^{T})^{−1}(D_{u_{1},..,u_{r}}f^{T})x^{∗}(t)

such that for almost all t ∈ [0, t_{f}], the instantaneous value u(t) of the control solves the following finite-dimensional optimization problem over all vectors v satisfying the constraints v ∈ Γ, ϕ_{i}(x(t), v) ≤ 0, i = 1, ..., r:

(∗)  max_{v} ⟨x^{∗}(t), f(x(t), v, t)⟩

where x(t) denotes the state trajectory corresponding to the control u(t), and ⟨·, ·⟩ is the Euclidean inner product.

To use the maximum principle, one solves (∗) to find the optimal control(s) u(t) in terms of x^{∗}(t) and x(t). Since x depends on u here, the optimization in (∗) yields u in feedback form. Thus, the minimum time problem is reduced to a two point boundary value problem consisting in solving the coupled systems of ordinary differential equations for x and x^{∗} subject to the fixed boundary conditions on the state x, with control u given as the (feedback) solution of the nested finitedimensional optimization problem (∗). Solving the optimization in closed form turns out to be difficult for WMR designs having constrained wheels.

We work in the body frame. Notice that H is affine in the \underline{u}_{i}, and the constraints are convex in the \underline{u}_{i}. The starred variables appearing in the pseudo-Hamiltonian are obtained by solving the differential system of adjoint dynamics. We will illustrate the application of the maximum principle in the case of a bicycle with one fully controllable wheel and one unsteered, undriven wheel. Our analysis generalizes in a straightforward way to WMRs not having any wheels which are simultaneously steered and undriven. Designs with steered undriven wheels may be dealt with also, but the analysis is more complicated, as the axle direction \underline{â}_{i} for the i-th wheel will involve the state in the discontinuous combination (k̂ × \underline{v} − Ω\underline{x}_{i})^{∧} if the wheel is undriven and steered.

Example. Consider the bicycle WMR design from the example in subsection 1.2. Observe that the constraint

(\underline{T C}_{r}^{us,ud})  ⟨\underline{u}_{r}, k̂ × \underline{â}_{r}⟩ = 0

merely states that the control for the rear wheel may be written in the form

\underline{u}_{r} = |\underline{u}_{r}|\underline{â}_{r}

for some scalar-valued function |\underline{u}_{r}|. By declaring \underline{u}_{r} to be of this form, we may consider the scalar function |\underline{u}_{r}| to be the effective control for the rear wheel, and we may eliminate the constraint \underline{T C}_{r}^{us,ud}. Also, for the present bicycle design we have

k̂ × \underline{x}_{f} = |\underline{x}_{f}|\underline{â}_{r}, k̂ × \underline{x}_{r} = −|\underline{x}_{r}|\underline{â}_{r}

Therefore, according to the maximum principle, any optimal control pair (\underline{u}_{f}, |\underline{u}_{r}|) must maximize the pseudo-Hamiltonian function

H = −α_{f} ⟨u_{f}, \frac{1}{m}v^{∗} + \frac{1}{I}Ω^{∗}|\underline{x}_{f}|\underline{â}_{r}⟩−α_{r}|\underline{u}_{r}| ⟨\underline{â}_{r}, \frac{1}{m}v^{∗} − \frac{1}{I}Ω^{∗}|\underline{x}_{r}|\underline{â}_{r}⟩+⟨x^{∗}, v⟩+θ^{∗}Ω

subject to the constraints

α_{f}(\frac{1}{m} − \frac{1}{I}|\underline{x}_{r}||\underline{x}_{f}|)⟨\underline{u}_{f}, \underline{â}_{r}⟩ + α_{r}(\frac{1}{m} + \frac{1}{I}|\underline{x}_{r}|^{2})|\underline{u}_{r}| = −Ω⟨k̂ × \underline{v}, \underline{â}_{r}⟩,

|\underline{u}_{f}| ≤ 1, ||\underline{u}_{r}|| ≤ 1.

In (\underline{u}_{f}, |\underline{u}_{r}|)-space, these latter norm constraints describe a bounded cylinder. The former (dynamic rolling) constraint defines a plane with normal vector given by

n = (\frac{1}{m} − \frac{1}{I}|\underline{x}_{r}||\underline{x}_{f}|)\underline{â}_{r} + (\frac{1}{m} + \frac{1}{I}|\underline{x}_{r}|^{2})\underline{ẑ}_{r},

where \underline{â}_{r} is now considered to be a vector in the \underline{u}_{f} plane, and \underline{ẑ} is the unit vector in the direction of the positive |\underline{u}_{r}| axis. The term −Ω⟨k̂ × \underline{v}, \underline{â}_{r}⟩ represents a parallel translation depending on the state (\underline{v}, Ω). Thus, the normal direction is fixed, but the plane is translated according to the state.

In this way, we arrive at the problem of maximizing the affine function H over the intersection of a (moving) plane and a bounded cylinder, i.e. over a truncated ellipse. This problem may be solved in closed form, yielding the optimal controls (\underline{u}_{f}, |\underline{u}_{r}|) explicitly, in terms of the state (\underline{v}, Ω) and the adjoint state (\underline{v}^{∗}, Ω^{∗}). The resulting expressions are somewhat cumbersome, though, so we won’t burden the reader with these details. Instead, let us qualitatively describe the optimization, in geometric terms. By projecting the gradient D_{u}H onto the plane of the ellipse, the problem is reduced to a planar one. A point p of the ellipse will satisfy the standard (Karush-Kuhn-Tucker) first order condition for a maximum, iff the entire ellipse lies to one side of the line passing through p and normal to the vector D_{u}H (which is constant in space), namely, the side lying in the direction in which H decreases. If the ellipse is not truncated, then each possible choice of a 2-vector value for D_{u}H determines a unique point p on the boundary of the ellipse in this way, unless D_{u}H is zero (i.e. unless the original three dimensional vector D_{u}H is normal to the plane of the ellipse). Moreover, a smoothly time-varying nonzero D_{u}H will produce a smoothly time-varying maximum point p on the boundary, i.e. the optimal controls will vary smoothly with time. However, for a truncated ellipse, each of the corner points p_{1}, p_{2} shown in the drawing will correspond to all values D_{u}H lying in some nontrivial interval of polar angles.

F IGURE NEEDED HERE

If we think of D_{u}H as being selected from ℝ^{2} at random, this means that the corner points will be selected with strictly positive probability. Furthermore, smoothly varying D_{u}H can now give rise to nonsmooth p; namely, if the moving p is “captured” by a corner, and if D_{u}H continues turning in the direction of the other corner, then it is possible that eventually, after staying at the first corner for a nontrivial interval of time, the mobile point p will jump to the second corner. Thus, we expect the optimal controls to exhibit discontinuous switching behavior, whenever the state is such that the parallel translation term −Ω⟨k̂ × \underline{v}, \underline{â}_{r}⟩ produces a truncated ellipse. For certain values of the parallel translation term, the ellipse will acquire four corners, and more complex switching behavior should be possible.

A similar analysis applies for more general WMR designs having only wheels which are either fully controllable or both undriven and unsteered. We would be interested in seeing whether numerical solutions of the two point boundary value problem obtained from the maximum principle for such designs, involving the nested convex optimization problem described above, actually exhibit the switching behavior as predicted by our qualitative analysis. At present, we have only calculated solutions numerically for the case of a tricycle with fully controllable wheels, however (see section 2.1).