CHAPTER 14
MOBILE ROBOTS

In this chapter we discuss the control of nonholonomic systems with a focus on the control of mobile robots. Mobile robots are becoming increasingly important in many applications, such as factory and warehouse automation, service and cleaning, household and medical assistance, military applications, agricultural and forestry applications, and search and rescue. Autonomous highway vehicles (cars and trucks) can also be considered mobile robots.

A distinguishing feature of mobile robots that impacts the control problem is the presence of nonholonomic constraints. Nonholonomic constraints arise in two primary ways.

1. In rolling without slipping constraints. For example, the translation and rotation of a wheel are not independent if the wheel rolls without slipping. Examples include:
   • A wheeled mobile robot, unicycle, automobile, or tractor/trailer
   • Manipulation of rigid objects, for example, fingers in rolling contact with an object
2. In systems where angular momentum is conserved. Examples include:
   • Satellites and space robots
   • Gymnastic, diving, and running robots

We will mainly focus in this chapter on the control of wheeled mobile robots, which are part of a particular class of systems known as driftless systems, which we define later. These systems are examples of underactuated systems that are not linearly controllable according to Definition 13.3 in Chapter 13.

We first discuss the notion of nonholonomic constraints and nonholonomic systems, which we introduced briefly in Chapter 6. Understanding the difference between holonomic and nonholomic constraints is fundamental to the understanding of every concept introduced in this chapter. We then discuss the problems of controllability and stabilizability for mobile robots and outline both open-loop and closed-loop control methods for the problems of set-point regulation and trajectory tracking. The main theoretical control results we will discuss are Chow’s theorem and Brockett’s theorem. Chow’s theorem gives necessary and sufficient conditions for a driftless system to be controllable. Brockett’s theorem gives a necessary condition for stabilization of such systems by smooth feedback control.

14.1 Nonholonomic Constraints

We introduced holonomic constraints in Chapter 6 when we derived the Euler–Lagrange equations of motion for rigid manipulators. We saw that interconnecting n rigid bodies, each of which independently has 6 degrees of freedom, with revolute or prismatic joints, introduces holonomic constraints that reduce the number of degrees of freedom from 6n to just n. Our treatment of force control in Chapter 10 dealt with unilateral constraints, which are imposed by contact between the robot and the environment. In this section we expand upon the notion of systems subject to constraints and discuss nonholonomic constraints.

Let (q₁, …, q_n) denote a coordinate representation of the configuration q in the configuration space of a given system. We recall the following definition from Section (6.1.2).

Definition 14.1. (Holonomic constraint).

(14.1)

As a result of this holonomic constraint, the configuration of the system, q = (q₁, …, q_n), is restricted to an n − 1 dimensional manifold in the configuration space defined by Equation (14.1).

Example 14.1.

(14.2)

We often have constraints that are expressed as functions of both the configurations and velocities as , where the function is continuously differentiable in both arguments. We will limit our discussion to so-called Pfaffian constraints, as defined next

Definition 14.2. (Pfaffian constraint).

(14.3)

Note that the constraint given by Equation (14.3) constrains the velocity vector at each q but does not necessarily constrain the configuration q to a lower dimensional manifold. Stated another way, at each configuration q, Equation (14.3) specifies the allowable velocities.

If the configuration q(t) is constrained by the holonomic constraint in Equation (14.1) for all time t ⩾ 0 it follows that

(14.4)

Thus we can say that a Pfaffian constraint is holonomic if there exists a function such that

This, in turn, implies the existence of a function γ(q), called an integrating factor such that

(14.5)

In this case, the constraint is referred to as integrable. Note that if the constraint is integrable, it follows, since h is continuously differentiable, that

This leads to the following necessary and sufficient conditions for a Pfaffian constraint to be integrable.

Proposition 14.1

(14.6)

Example 14.2.

(14.7)

which is of the form with w = ( − q₃, 1, 0). A simple calculation, using Equation (14.6), shows that

It is easy to see from the above that the only possible solution is γ = 0 (Problem 14–1). Therefore the constraint is nonholonomic.

Multiple Constraints

Suppose now that we have k ⩾ 2 Pfaffian constraints

(14.8)

where each covector w_i(q) is given by w_i(q) = (w_i1(q), …, w_in(q)).

Example 14.3.

We note that checking the holonomy of each constraint separately does not give us sufficient information about the constraints on the system configuration. Consider the two Pfaffian constraints

(14.9)

(14.10)

One can check that individually these constraints are nonholonomic. In fact, we showed the nonholonomy of the constraint given by Equation (14.9) in Example 14.2 above. Note, however, that the linear combination q₁w₁ + w₂ satisfies

(14.11)

which can be integrated to and (Problem 14–2). Therefore, the constraints are holonomic. In fact, it will follow from Theorem 14.1 below that any set of n − 1 Pfaffian constraints in is holonomic.

We can express a set of multiple Pfaffian constraints, as in Equation (14.8) as a matrix equation

(14.12)

where A is the k × n matrix

(14.13)

Note that means that the velocity lies in the null space of A(q) for each q. If the rank of A(q) is equal to k, implying that the constraints are independent, then there will be an m = n − k-dimensional null space of A. Let {g₁(q), …, g_m(q)} be a basis for the null space of A(q). Then each velocity vector satisfying Equation (14.8) can be expressed as

(14.14)

for suitable coefficients u₁, …, u_m. Equation (14.14) specifies the allowable velocity vectors as a linear combination of the basis vectors g₁, …, g_m.

Example 14.4.

Consider again the k = 2 constraints in Equations (14.9)–(14.10) on . We can write these as where

It is now easy to show by direct calculation that

spans the one-dimensional null space of A.

Remark 14.1.

Equation (14.14) is called a driftless system because when the control inputs u₁, …, u_m are zero. In many systems of interest the coefficients u_i in Equation (14.14) have the interpretation of control inputs. We will use this model in later sections for control design.

14.2 Involutivity and Holonomy

In this section we refine the notion of nonholonomic constraints that will be useful later when we discuss controllability of nonholonomic systems and Chow’s theorem.

Definition 14.3.

(14.15)

Theroem 14.1.

Proof: Using the Lie derivative notation from Chapter 12, we can say that there exist functions h_i, …, h_k such that

(14.16)

if and only if the constraints are holonomic. From the Frobenius theorem the set of partial differential equations

has a solution for h₁, …, h_k if and only if the distribution Δ is involutive, and the result follows.

Remark 14.2.

Example 14.5.

(14.17)

(14.18)

Filtrations

Suppose now that the distribution Δ is not involutive. Then there is at least one pair of vector fields g_i and g_j in Δ such that the Lie Bracket [g_i, g_j] does not belong to Δ. Thus, with Δ₁ = Δ =  span{g₁, …, g_m}, we define Δ₂ as the distribution

where

where [g_i, g_j] is the Lie Bracket of the vector fields g_i and g_j. The distribution Δ₂ is spanned by all vectors g_i ∈ Δ₁ together with all possible vectors generated by taking Lie Brackets of pairs of vectors in Δ₁. Thus, if Δ₂ = Δ₁ it simply means that the original distribution Δ is involutive.

If Δ₁ is not involutive, then Δ₂ ≠ Δ₁ and its dimension is strictly larger than the dimension of Δ₁. We can continue this process and define a sequence of distributions Δ_i, with Δ₁ = Δ according to

where

This sequence of distributions {Δ_i} is called a filtration associated with Δ.

Definition 14.4.

A filtration is said to be regular in a neighborhood U of q₀ if

In other words, Δ_i has constant rank for all q.

If the filtration is regular then the dimension of Δ_i is greater than or equal to the dimension of Δ_{i − 1} and cannot be greater than n, the dimension of the configuration space itself. If at some point there exists κ such that  rank Δ_{κ + 1} =  rank Δ_κ then the above construction terminates. This leads us to

Definition 14.5.

Let {Δ_i} be a regular filtration associated with a distribution Δ. The smallest integer κ such rank Δ_{κ + 1} = rank Δ_κ is called the degree of nonholonomy of the distribution Δ.

Note that m ⩽  rank Δ_κ ⩽ n and 1 ⩽ κ ⩽ n − m + 1. The degree of nonholonomy is related to the difficulty (or ease) of the motion planning and control problems. Based on the above derivation we can refine our definition of holonomic constraints as follows

Definition 14.6.

The set of k Pfaffian constraints, given by Equation (14.8), is

1. Holonomic if κ = 1 or, in other words, if the dimension  dim Δ₁ = n
2. Partially Nonholonomic if m <  dim Δ_κ < n, i.e. 1 < κ < n − m + 1
3. Completely Nonholonomic if dim Δ_κ = n, i.e. κ = n − m + 1

The distribution Δ_κ is called the involutive closure of Δ, denoted , and by construction, is the smallest involutive distribution containing Δ, in the sense that if Δ′ is an involutive distribution such that Δ⊂Δ′, then .

14.3 Examples of Nonholonomic Systems

In this section we give examples of systems subject to nonholonomic constraints.

Example 14.6. (The Rolling Disk).

(14.19)

where r is the radius of the wheel. These constraints can be written as Pfaffian constraints in the form of Equation (14.3) with q = [x, y, θ, rϕ]^T and

Since the dimension of the configuration space is n = 4 and there are two constraint equations, a basis for Ω^⊥ will consist of two function g₁, g₂ orthogonal to w₁, w₂. It is easy to see that

(14.20)

(14.21)

We can now check to see if rolling without slipping constraints on the disk are holonomic or nonholonomic using Theorem 14.1. Computing the Lie brackets (Problem 14–3)

(14.22)

we see that

Therefore . Thus the degree of nonholonomy is 3 and the constraints are completely nonholonomic.

Remark 14.3.

If we do not care about the orientation of the disk about the horizontal axis, i.e., if we ignore the last equation , then the constraints in Equation (14.19) can be reduced to the single constraint

and we have a reduced-order model

(14.23)

Example 14.7. (The Kinematic Car).

(14.24)

(14.25)

where the covectors w₁ and w₂ are

It is now straightforward to find vectors g₁ and g₂ orthogonal to w₁ and w₂ and write the corresponding system in the form of Equation (14.14). For example, we may take

(14.26)

Computing the Lie Brackets [g₁, g₂] and [g₂, g₃] gives

We have that

is involutive of rank 4. Hence the system has degree of nonholonomy κ = 3.

Example 14.8. (A Hopping Robot).

Consider the hopping robot in Figure 14.4. The configuration of this robot is defined by q = (ψ, ℓ, θ), where

ψ	=	the leg angle
θ	=	the body angle
ℓ	=	the leg extension

During its flight phase the hopping robot’s angular momentum is conserved. Letting I and m denote the body moment of inertia and leg mass, respectively, conservation of angular momentum leads to the expression

(14.27)

(14.28)

(14.29)

Example 14.9. (Differential Drive Robot).

(14.30)

where v and ω are the linear and angular velocities of the chassis, respectively. Assume that each wheel has radius r and that the distance between the wheels is 2b. Assuming no wheel slipping, we can relate the chassis velocities v and ω to the left and right wheel velocities and as

Solving for v and ω gives

Note that actuating both wheels in the same direction with speed results in a pure translation of the chassis with speed while actuating both wheels in opposite directions with speed results in a pure rotation with angular speed .

Typically the wheels are actuated by DC-electric motors. Setting τ_R and τ_L as the left and right motor torques, respectively, we can express the dynamics of the DDR as

where m is the total mass of the DDR and J is its moment of inertia about the center of mass, which we can take as the midpoint of the wheel axle. With

and

we have

where and . If we include damping terms we can write the dynamic equations relating the motor torque to wheel acceleration as

(14.31)

14.4 Dynamic Extension

Dynamic models for nonholonomic systems can be obtained in a straightforward way as an extension of the previous kinematic models. These extended models typically take the form

(14.32)

(14.33)

Note that the system in Equations (14.32)–(14.33) is now a system with drift of the form

The above model can be derived from the Euler–Lagrange equations in the form

(14.34)

(14.35)

where is defined using the nonholonomic Pfaffian constraints and λ is a vector of Lagrange multipliers. This model relates to the kinematic model

(14.36)

if we take, as before, g₁, …, g_m as a basis for the null space of A(q). Differentiating Equation (14.36) gives

(14.37)

Since G^T(q)A^T(q) = 0 by choice of basis vectors g_i, if we substitute the expression for from Equation (14.37) in into the Euler–Lagrange equations and multiply by G^T(q) we obtain

(14.38)

where

(14.39)

Assuming G^T(q)B(q) is invertible, the control input τ can be chosen as a feedback linearizing control

(14.40)

to obtain

(14.41)

Note that τ depends on the control input u since . We can compute u from Equation (14.36) as

where G† is the right pseudoinverse of G, and substitute this expression into Equation (14.39) to complete the computation of the control τ.

Example 14.10.

For the unicycle model,

let τ₁ be the driving force and τ₂ be the steering torque so that

where m is the unicycle mass and J the moment of inertia about the vertical axis. Then a simple dynamic extension is given by

with ν₁ = τ₁/m and ν₂ = τ₂/J. To see how this model is obtained from the general theory, we can compute, using the above equations,

which we can write as

where the second equation above describes the nonholonomic constraint. Thus, with A and G given by

the above equations are of the form

and a straightforward calculation (Problem 14–10) gives

and therefore we obtain

as above.

14.5 Controllability of Driftless Systems

In this section we consider the controllability of driftless systems of the form Equation (14.14). There are several notions of controllability for general nonlinear systems, all of which are equivalent for linear systems. The examples of wheeled mobile robots that we consider in this chapter all satisfy the strongest notion of controllability defined next and so we will not have need for other (weaker) notions of controllability.

We assume that the vector fields g₁(q), …, g_m(q) are smooth, complete,¹ and linearly independent at each . Under these conditions we can state the following:

Definition 14.7.

The driftless nonlinear system in Equation (14.14) is controllable if, for any q₀ and q₁ , there exists a time T > 0 and a control input

such that the solution q(t) of Equation (14.14) satisfies q(0) = q₀ and q(T) = q₁.

Let us digress for a moment to gain an intuitive understanding of the notion of nonlinear controllability and its relation to nonholonomy. Consider the familiar problem of parking a car. Referring to Figure (14.6) the desired motion is to move the car laterally, which is prohibited by the rolling constraints on the wheels. However, by sequentially applying the two control inputs, namely 1) turning the wheels left and right, and 2) driving forward and backward, one is able to park the car. With this example in mind, let the system

(14.42)

be defined by the two vector fields g₁ and g₂ and consider the following thought experiment consisting of sequentially applying the control u₁ and u₂:

• Set u₁ = 1 and u₂ = 0 and follow the vector field g₁ for a time t = ε.
• Then set u₁ = 0 and u₂ = 1 to follow the vector field g₂ for time ε < t ⩽ 2ε.
• Reverse the above by setting u₁ = −1, u₂ = 0 for time 2ε < t ⩽ 3ε.
• Finally, set u₂ = −1 and u₁ = 0 for time 3ε < t ⩽ 4ε.

Denote by ϕ^g_t(q₀) the state of the system at time t starting at q₀ at time t = 0. The function ϕ is called the flow of the vector field g. With this notation, the state q(4ε) that is reached after the above thought experiment is given by

It is shown in most basic texts in differential geometry using a Taylor series expansion of the flow that the resulting state reached after the four sequential moves is given by

In other words, the motion is equivalent, for small ε, to moving in the Lie bracket direction [g₁, g₂]. We will show later that the lateral motion of the car is precisely the direction given by the Lie bracket of the ‘turning’ and ‘driving’ input vectors.

The above discussion suggests that the possible directions of motion, i.e., the velocity directions, include not only the directions given by g₁, …, g_m but also motions in the direction [g_i, g_j], [g_k, [g_i, g_j]], and so on. Thus, it is important to know if the vectors [g_i, g_j] are simply linear combinations of the original vectors g₁, …, g_m or if they are vectors defining new and independent directions.

The next result, known as Chow’s theorem, formalizes this notion and gives a necessary and sufficient condition for a system given by Equation (14.43) to be controllable.

Theroem 14.2. (Chow).

(14.43)

The condition is called the controllability rank condition. The involutive closure of Δ is also called the control Lie algebra for the system in Equation (14.43).

Example 14.11.

Consider the following system on

with three states q₁, q₂, q₃, and two control inputs, u₁ and u₂. It is easy to see that the distribution Δ =  span{g₁, g₂} has rank two for all values of . The Lie bracket [g₁, g₂] is

and, therefore, we have

for all values of . Therefore, by Chow’s theorem, the system is controllable on .

Remark 14.4.

14.6 Motion Planning

We have given several examples of nonholonomic mobile robots that are controllable according to Chow’s theorem. Chow’s theorem tells us when a driftless system is controllable but does not tell us how to find the control input to steer the system from a given initial state x₀ to a desired final state x₁. In this section we present several ideas for planning feasible trajectories for these systems.

14.6.1 Conversion to Chained Forms

We have seen the usefulness of transforming systems into canonical or normal forms through the use of nonlinear coordinate transformations and nonlinear feedback, as in the feedback linearization in Chapter 12 and partial feedback linearization in Chapter 13. In this section we introduce the so-called chained form which gives a useful transformation of the kinematics of many nonholonomic systems. We limit the discussion to systems with two inputs, such as the unicycle, DDR, and kinematic car.

Example 14.12.

To motivate the subsequent development, consider again the unicycle model

Define a coordinate transformation

Note that the coordinates (z₂, z₃) represent a rotational transformation of (x, y) through the angle θ. In terms of the transformed coordinates, z₁, z₂, and z₃, the input transformation

results in the system (Problem 14–5)

(14.44)

which is referred to as a (2, 3)-chained system since it has two inputs and three states.

We note that, like the unicycle model it derives from, Equation (14.44) defines a driftless nonlinear system, and can be shown to be controllable using Chow’s theorem (Problem 14–12).

In the system of equations (14.44) the variables z₁ and z₂ are called base variables and can be independently controlled with inputs v₁ and v₂, respectively. The third coordinate z₃ is called a fiber variable and evolves according to the input z₂v₁ and therefore cannot be independently controlled.

Steering Using Sinusoids

The chain form representation is useful for deriving trajectories in configuration space using several methods. One such method is to use sinusoidal functions at different frequencies for the inputs v₁ and v₂ to move the shape variables through periodic motions. This will move the fiber variables to new values that are functions of the amplitudes and frequencies of the input functions. The motion planning problem then becomes one of choosing appropriate values for the amplitudes and frequencies of v₁ and v₂. We illustrate this idea below for the case of the unicycle model.

Example 14.13.

Consider the chained-form system derived above from the unicycle model

(14.45)

and let us choose the control inputs v₁ and v₂ to minimize the quadratic cost function

To solve this problem we form the Hamiltonian function

(14.46)

where the Lagrange multiplier vector λ = (λ₁, λ₂, λ₃) satisfies the adjoint equations . Since the Hamiltonian is independent of z₁ and z₃, the adjoint equations are easily computed as

(14.47)

The necessary condition for optimality, , leads to

(14.48)

(14.49)

from which we get

(14.50)

Differentiating v₁ and v₂ in Equation (14.50) and using Equation (14.47) yields

(14.51)

(14.52)

which has the general solution

(14.53)

(14.54)

Thus, we have shown that the control for the chained-form system that minimizes the cost function (1.46) consists of sinusoidal inputs.

This result leads to the following motion planning strategy for the above system:

• First, use any method to move the base variables z₁ and z₂ to their desired final values, z^d₁ and z^d₂, respectively. Since the base variables are independently controlled, this step is simple and should be completed in finite time t₁.
• Next, execute a periodic motion of the base variables over one period T. This motion will bring the base variables back to their desired final value and move the fiber variable z₃ from z₃(t₁) to z₃(t₁ + T). Choose the periodic motion of the base variables so that z₃(t₁ + T) = z^d₃, the desired value for z₃.

The strategy is suboptimal since the control of the shape variables in the first step is not necessarily optimal. If we now choose the controls v₁ and v₂ as

it is easily shown (Problem 14–8) that after 2π/ω seconds, z₁ and z₂ return to their initial values whereas the change in z₃ is abπ/ω².

Example 14.14.

Suppose that we wish to move the unicycle from the origin (z₁, z₂, z₃) = (0, 0, 0) to (x₁, x₂, x₃) = (0, 0, 10) in two seconds. Using the above controls with a = π, ω = π, b = 10 results in the response shown in Figure 14.8.

Chained Form for Higher Dimensional Systems

We can generalize the previous idea of conversion to chained form to higher dimensional (2, n)-chained form systems as follows.

Definition 14.8.

A (2, n) chained-form is a driftless system with 2 inputs, v₁ and v₂, and n configuration variables z₁, …, z_n in the form

(14.55)

In this case, z₁ and z₂ are the base variables and z₃, …, z_n are the fiber variables.

The system in Equation (14.55) can be written in the form of a driftless system as

where

(14.56)

It is not too difficult to show that the chained-form system, Equation (14.55), is controllable according to Chow’s theorem. In order to see this, one can show by induction (Problem 14–6) that

where the non-zero entry (−1)^k appears in the k + 2 position. It follows by direct calculation that the distribution

has dimension n and controllability follows from Chow’s theorem.

We state without proof a sufficient condition for the two-input case for conversion to chained form. Specifically, given a driftless system of the form

(14.57)

there exist a diffeomorphic change of coordinates and a control input v = β(u) that transforms Equation (14.57) into the (2, n) chain form Equation (14.55) if the following distributions are constant rank and involutive

(14.58)

and there exists a function h₁(q) such that

(14.59)

and a function h₂, independent of h₁, such that

(14.60)

Note that Equations (14.59) and (14.60) are a set of partial differential equations that must be solved to find h₁ and h₂. The required change of coordinates in the state and control inputs are then given by

Example 14.15.

We can apply the above transformation to the kinematic car

(14.61)

It is straightforward (Problem 14–7) to compute the distributions Δ₀, Δ₁, and Δ₂ and verify that they are of constant rank and involutive. With h₁ = q₁ and h₂ = q₂ the state transformation

and input transformation

yields the required (2, 4) chain system.

For the (2, n) chained-form system, Equation (14.55), a similar procedure to the above can be applied to sequentially move each variable z_k to its desired value as follows:

• First choose v₁ and v₂ to move the base variables z₁ and z₂ to their desired values in time t₁. As before, the fiber variables z₃, …, z_n will move to new values z₃(t₁), …, z_n(t₁).
• For 3 ⩽ k ⩽ n, steer z_k to its desired value z^d_k with controls
  
  over one period, T = 2π/ω, where a and b are chosen to satisfy
  

Carrying out the sequential steering controls moves the chained-form system from z₀ to z^d in time t = t₁ + (n − 2)T.

Example 14.16.

Consider the (2, 4) chained-form system

and suppose we wish to move the states from (0, 0, 0, 0) to (0, 0, 10, 15) in 4 seconds. In the first stage we take v₁ and v₂ as in the previous example as

After 2 seconds, z₃ reaches its desired value of 10 and z₄(2) = −10. In the second stage we take

with . With z^d₄ = 15 and a = ω = π, this gives b = 100. The responses are shown in Figure 14.9.

14.6.2 Differential Flatness

In this section we introduce the notion of differential flatness and its application to the motion planning and control problems. The property of flatness simplifies the process of determining feasible trajectories for mechanical systems. This is particularly useful for underactuated systems and systems with nonholonomic constraints since, as we have seen, not all trajectories are feasible. It turns out that all of the systems considered in the chapter, indeed all driftless systems that can be converted to chained form, are differentially flat.

The basic idea of differential flatness is, given a system with n states and m inputs, find a set of m outputs so that the states and inputs can be expressed as functions of these m outputs and their derivatives. More precisely, we have

Definition 14.9.

The nonlinear system

is said to be differentially flat if there exists an output y = (y₁, …, y_m), with the same dimension as the control input, satisfying the following conditions:

• There exists a nonnegative integer α such for each i = 1, …, m, for some smooth functions h₁, …h_m.
• There are smooth functions, ϕ, and ψ, and a nonnegative integer r ⩾ 0 such that
  
• The components y_i of y are differentially independent in the sense that they satisfy no differential equation of the form .

The components of the vector y in the above definition are called flat outputs. In other words, once a suitable set of flat outputs is identified, the state x and input u of the system are completely determined as functions of the flat outputs and their derivatives. Specifically, for every sufficiently differentiable curve t → y(t), there corresponds a trajectory

The proof of this property is beyond the scope of this text. In the control field, the notion of differential flatness grew out of work on static and dynamic feedback linearization and is intimately related to these latter concepts.

Remark 14.5.

It is routine to show that any system that can be transformed to the Brunovsky form in Equation (12.35) is flat. This includes controllable linear systems as well as nonlinear systems that are feedback linearizable (Problem 14–13).

For the problem of trajectory tracking, suppose that we want to find a feasible trajectory with initial and final conditions given as (t₀, x(t₀), u(t₀)), and (t_f, x(t_f), u(t_f)), respectively. Then, we can calculate corresponding values and for the flat outputs with s ⩽ r. We can then interpolate a smooth curve t → y(t) between these initial and final values, which then determines the trajectory t → (x(t), u(t)) between t₀ and t_f. By construction, the computed trajectory automatically satisfies the system equations and hence any constraints, nonholonomic or otherwise.

Example 14.17.

For the DDR system, a set of flat outputs is given by y₁ = x, y₂ = y. The remaining variables are then

Suppose that we want to move the DDR from configuration (x, y, θ) = (0, 0, 0) at time t₀ = 0 to (x, y, θ) = (10, 10, 0) at time t_f = 1. Then, in terms of the flat outputs, the constraints are

where the last two constraints on indicate that the orientation θ of the DDR is horizontal at the beginning and end of the trajectory. Since there are two constraints on y₁ and four constraints on y₂, we can choose reference trajectories as polynomials of the form

Solving for the polynomial coefficients using the given constraints yields

Figure 14.10 shows the resulting trajectory and control inputs.

In general, determining when a given system is flat and finding a set of flat outputs is difficult. However, it is known that the (2, n) chain-form system is always flat with flat outputs y₁ = z₁ and y₂ = z_n.

Example 14.18.

For the chain-form system below, a set of flat outputs

is y₁ = z₁ and y₂ = z₃. Then it is straightforward to show that the remaining variables z₂, v₁, v₂ are given by

14.7 Feedback Control of Driftless Systems

In this section we consider the problem of finding feedback control laws, as opposed to the purely time-based, open-loop controllers that we derived in the previous section. We consider both the problem of pose regulation, i.e., point-to-point motion, and the problem of trajectory tracking. The first result below, known as Brockett’s theorem, gives a necessary condition for the existence of a smooth stabilizing control.

14.7.1 Stabilizability

Theroem 14.3. (Brockett).

Given the nonlinear system

(14.62)

with f(x, u) continuously differentiable and f(0, 0) = 0, suppose that there exists a continuously differentiable (smooth) feedback control law u = u(x) that makes the origin x = 0 asymptotically stable. Then the image of the map f contains a neighborhood of the origin.

Example 14.19.

For the system

It is easily seen that, given any a ≠ 0, there does not exist (x, u) satisfying

Thus, the image (range) of the function f does not contain a neighborhood of the origin and so there does not exist a smooth C¹ feedback control law u(x) that stabilizes the origin.

Most nonholonomic systems, including the examples considered in this chapter, do not satisfy Brockett’s necessary condition for the existence of a smooth stabilizing feedback control. As a result, Brockett’s theorem has motivated a great deal of research into finding alternative control schemes for more general nonholonomic systems. These include

• Open-loop control laws
• Time-varying control laws u = u(x, t)
• Discontinuous and switching control laws

Example 14.20.

Consider the unicycle system. It is easy to see that for any initial condition x₀, y₀, θ₀, the following switching control scheme drives the system to the origin, i.e. is a stabilizing control:

1. Rotate the unicycle in place to line up the orientation with the origin
2. Translate until (x, y) = (0, 0)
3. Rotate in place until θ = 0

We note that if the nonlinear system is linearly controllable, then there is always a smooth feedback control law, in fact, of the form u = −kx, that stabilizes the origin locally.

Example 14.21.

Consider the Acrobot model

and suppose that g₁ = 0. From Proposition (13.1) we know that this system is not linearly controllable about the origin. It is left as an exercise (Problem 14–9) to show that there is no smooth feedback control that stabilizes the origin.

14.7.2 Nonsmooth Control

One way to circumvent the obstruction due to Brockett’s theorem is to use discontinuous or switching control.

Sliding-Mode Control

The possibility of using discontinuous control to circumvent the obstruction imposed by Brockett’s theorem suggests the application of sliding-mode control. Consider the chain-form system

Define σ = z₃ − z₁z₂/2 and let the controls v₁ and v₂ be given as

(14.63)

(14.64)

We want the manifold σ = 0 to serve as a sliding surface for the system trajectory. Let V be given as

(14.65)

Then

(14.66)

(14.67)

Thus, V(t) = V(0)e^{− 2t} → 0 as t → ∞ and therefore z₁(t) and z₂(t) converge asymptotically to zero. With the given choice of σ a straightforward calculation shows

(14.68)

Therefore, |σ(t)| is nonincreasing. If σ(t) reaches the origin at some time t = T it will remain at the origin and slide along the surface z₃ = z₁z₂/2 toward the origin. Figures (14.11) and (14.12) shows a simulation of this controller.

Dynamic Extension

For systems like the unicycle or DDR, it may be difficult to implement such a sliding-mode controller that requires rapid switching of the velocities v and ω. For this reason we may consider the dynamic extension

For example, u₁ and u₂ could represent the torque inputs, which we can take as

where we choose

as reference trajectories for v₁ and v₂. This will have the effect of smoothing the control signals and is a more practical control from the standpoint of implementation. Figures (14.13) and (14.14) show the responses with the modified controls.

14.7.3 Trajectory Tracking

Lyapunov Design

In this section we illustrate the use of Lyapunov-like design for pose regulation of the DDR robot. Considering the DDR model

(14.69)

let us first change to polar coordinates (r, ϕ) by setting

and define β = ϕ − θ. In terms of coordinates r, β, and ϕ a straightforward computation gives

(14.70)

Note that Equation (14.70) is only valid on R³ − {0} because of the singularity at r = 0. Thus, the assumptions of Brockett’s prohibiting the design of a smooth stabilizing control law do not hold. Define a candidate Lyapunov function

(14.71)

Then, satisfies

If we set the controls v and ω as

(14.72)

(14.73)

a straightforward calculation shows that

(14.74)

Due to the singularity at r = 0, some additional care must be taken to conclude that the trajectory of the system converges asymptotically to the origin. We note that Equations (14.71) and (14.74) together imply that all solutions are bounded and that asymptotically. Moreover, r does not reach zero in finite time due to the boundedness of β and the expression for . An application of Barbalat’s Lemma (see Appendix C) now implies that all trajectories asymptotically converge to the origin.

Example 14.22.

Suppose we want to move the DDR from the initial pose (x, y, θ) = (− 1, 1, 3π/4) to the origin. The corresponding initial conditions for the transformed coordinates (r, β, ϕ) are . Figure 14.15 shows the path computed using the above procedure.

14.7.4 Feedback Linearization

In this section we present the application of feedback linearization for the problem of trajectory tracking for mobile robots. Thus, instead of pose regulation, we want the system configuration to follow a time-varying trajectory. With regard to static feedback linearization, the driftless system

(14.75)

is feedback linearizable in a neighborhood of q₀ if and only if  rank G(q) = n in . This requires the system to be fully actuated, i.e. m = n, in which case the distribution

(14.76)

is trivially involutive and a feedback linearizing control is given by

(14.77)

In the more interesting underactuated case m < n we may apply the method of partial or input/output linearization to try to linearize m degrees of freedom. In this case, the problem is to determine a suitable output with respect to which a linear input/output relation can be established.

Example 14.23.

Consider again the differential drive robot

(14.78)

and suppose that we choose the outputs

(14.79)

(14.80)

with d ≠ 0. The outputs are the coordinates of the point d shown in Figure 14.16. We can take d positive if it is located ahead of the wheel axle and negative otherwise. Differentiating the outputs gives

(14.81)

The matrix A is invertible for d ≠ 0 and the input transformation

(14.82)

results in

Using the expression for ω from Equation (14.82) yields the 3-dimensional normal form

(14.83)

Thus, given a desired output trajectory (y^d₁(t), y₂^d(t)), which is really a desired trajectory for the point d to follow, the control input

(14.84)

(14.85)

results in exponential tracking of the output trajectory. Note that the zero dynamics are given by the equation

(14.86)

Figure 14.17 shows a simulation of the above controller with reference trajectory

The initial conditions were chosen as (0, 0, 0) with d = 0.1.

Dynamic Feedback Linearization

The above method of input/output linearization fails for d = 0. To have the (x, y) coordinates track desired reference trajectories, we will use the method of dynamic feedback linearization, which we have so far not considered in any applications. To understand this design method, we consider the differential drive robot

with output equations

Differentiating the outputs gives

(14.87)

(14.88)

Differentiating the outputs a second time would result in the derivative of v appearing in the equations, which we want to avoid. Therefore, we let v be represented as a new state variable ξ = v. Then , the linear acceleration. Now, if we differentiate Equations (14.87)–(14.88) with respect to time we get

Thus, we can take

which results in

as long as ξ ≠ 0. Given a desired output trajectory (y^d₁(t), y₂^d(t)) we can set

and the tracking errors e_i = y_i − y^d_i satisfy

and exponentially convergence to zero. The resulting dynamic compensator is given by

(14.89)

Note that the angular velocity ω in Equation (14.89) is unbounded as the linear velocity ξ tends to zero. This means that the dynamic feedback linearization approach requires that the robot never stop moving and, hence, this method cannot be used for pose regulation.

14.8 Chapter Summary

This chapter provided an introduction to the control of nonholonomic systems with particular emphasis on wheeled mobile robots. We provided examples and case studies for several models including the unicycle, differential drive robot and a car-like robot. All of these systems are modeled as driftless systems of the form

with n configuration variables and m < n control inputs.

Problems

1. Verify that the only solution for γ to the equations in Example 14.41 is γ = 0.
2. Verify that the constraint in Equation (14.11) can be integrated to and .
3. Fill in the details in Example 14.1 showing that the constraints are nonholonomic.
4. Fill in the details in Example 14.6 necessary to derive the vector fields g₁ and g₂ and show that the constraints are nonholonomic.
5. Verify the calculations given in Example 14.7 that transform the DDR model into chain form.
6. Show by induction that the (2, n) chained-form system in Equation (14.55) is controllable.
7. Compute the distributions Δ₀, Δ₁, and Δ₂ in Example 14.15 and verify that they are of constant rank and involutive.
8. Carry out the details to verify the claim in Example 14.21 that the variables z₁ and z₂ return to their initial values after 2π/ω seconds and the change in z₃ is abπ/ω².
9. Show, using Brockett’s theorem, that there does not exist a smooth stabilizing feedback control for the Acrobot model in Example 14.21 if g₁ = 0.
10. Verify the calculations in Example 14.10.
11. Consider the two-link planar RR robot
    
    
    Investigate the existence of a smooth stabilizing control law under various scenarios regarding u₁, u₂, g₁, g₂.
12. Use Chow’s theorem to show that the system defined by Equation (14.44) is controllable.
13. Let be a single-input nonlinear system and suppose that the system is feedback linearizable with a nonlinear change of coordinates and nonlinear feedback. Show that the system is differentially flat.

Notes and References

A more complete treatment of the control of nonholonomic systems, including mobile robots, can be found in [118] and [32]. An excellent introduction to the topics here is [100], which gives examples of underactuated planar manipulators in addition to wheeled mobile robots. Another good tutorial on the topic of nonholonomic control is [80]. Additional references on nonholonomic behavior and control of planar manipulators are [130, 101, 102].

Our treatment of systems in chained form follows closely the development in [118] and [100]. The treatment of steering using sinusoids follows [121] and [118]. A proof of Brockett’s theorem is in [18]. An excellent treatment of switching control is [95]. Additional details on differential flatness and its application to nonholonomic motion planning is in [120, 141]. The Lyapunov-like approach to control of the unicycle using polar coordinates is taken from [3]. More details on the concept of dynamic extension can be found in [13, 80]. Dynamic feedback linearization is treated in [99, 101, 22].