# Nonholonomic motion planning: steering using sinusoids

## Summary (3 min read)

### I. INTRODUCTION

- OTION planning for robots has a rich history.
- The authors interests in motion planning are not along the lines of the aforementioned approaches, but are complemen-Manuscript received May 22, 1991; revised April 5, 1992 tary: they involve motion planning in the presence of nonholonomic or nonintegrable constraints.
- If an object is twirled through a cyclic motion that returns the object to its initial position and orientation, and the fingers roll without slipping on the surface of the object, the fingers do not necessarily return to their initial configurations.
- Nonholonomic constraints arise either from the nature of the controls that can be physically applied to the system or from conservation laws which apply to the system.
- In Section 111, using some outstanding results of Brockett on optimal steering of certain classes of systems as motivation [6], the authors discuss the use of sinusoidal inputs for steering systems of first order, i.e., systems where controllability is achieved after just one level of Lie brackets of the input vector fields.

### 11. MATHEMATICAL PRELIMINARIES

- This section describes the notation to be used throughout the paper and collects a variety of results from nonlinear control theory and Lie algebras which will be used in the sequel.
- For basic definitions and concepts in differential geometry, see Boothby [5] or Spivak [48] .
- A good introduction to nonlinear control theory which includes many of the necessary differential geometric concepts can be found in Isidori [21] or Nijmeijer and van der Schaft M11.

### A. Nonlinear Control Theory

- The net motion consists of flowing along g,,g,, -gl, -g, for time E and can be shown to satisfy (3) Thus, the Lie bracket is the infinitesimal motion that results from flowing around a square defined by two tangent vectors.
- Roughly speaking, Chow's theorem states that if the authors can move in every direction using Lie bracket motions (possibly of higher order than one), then the system is controllable.

### C. Examples of Nonholonomic Systems

- The robot consists of a leg which can both rotate and extend.
- The configuration of the mechanism is given by the angle of the body and the angle and length of the leg.

### These algorithms.

- This robot consists of a body with an actuated leg that can rotate and extend; the "constraint" on the system is conservation of angular momentum.
- For simplicity, the authors take the body mass to be one and concentrate the mass of the leg, m,, at the foot.
- The constraints for the front and rear wheels are formed by writing the sideways velocity of the wheels:.
- The wheels of the individual trailers are aligned with the body of the trailer.

### A. First-Order Systems

- This problem is related to finding the geodesics associated with a singular Riemannian metric (Camot-Caratheodory metric).
- To solve the problem, Brockett considers a class of systems which have a special canonical form.
- Using this as justification, the authors attempt to use their proposed algorithm to steer the full nonlinear system.
- Since the authors control the @ and 1 states directly, they first steer them to their desired values.

### B. Second-Order Systems

- The authors next consider systems in which the first level of bracketing is not enough to span R".
- The authors begin by trying to extend the previous canonical form to the next higher level of bracketing.
- Consider a system which can be expressed as EQUATION Because Jacobi's identity imposes relations between certain brackets, not all xlJk combinations are permissible if the system (10) is to be completely controllable.
- Constructing the Lagrangian (with the same integral cost function) and substituting into the Euler-Lagrange equations does not in general result in a constant set of Lagrange multipliers, although Brockett and Dai have shown that for m = 2 the optimal inputs are elliptic functions [9] .
- The authors can extend and apply their previous algorithm as follows.

### Iv. CHAINED SYSTEMS

- The authors now study more general examples of nonholonomic systems and investigate the use of sinusoids for steering such systems.
- As in the previous section, the authors try to generate canonical classes of higher order systems, i.e., systems where more than one level of Lie brackets is needed to span the tangent space to the configuration space.
- The authors give sufficient conditions under which systems can be transformed into a chained form and show the procedure applied to several illustrative examples.

### A. Maximum Growth Canonical Systems

- Hall basis, it is possible to construct vector fields which have maximum growth; at each level of bracketing the dimension of the filtration grows by the maximum possible amount.
- Hall basis elements, the resulting set of vector fields is linearly independent.
- Thus, the authors can associate with each such basis element a well-defined vector a E B" which indicates the number of times each basis element occurs in the expansion (12); i.e., ai(k) is the number of times B, appears in the expansion for Bi.
- Then nents in the derivatives fo the dynamic system: Example 8: Consider the two input example given previously, but with order of nilpotency 4 instead of 5. sions of the canonical forms the authors have seen for degree of nonholonomy and 3.
- It may still be possible to steer the system using combinations of sinusoids at different frequencies for each input or using more complicated periodic functions (such as elliptic functions, see [91).

### B. Chained Systems

- Rather than explore the use of more complicated inputs for steering nonholonomic systems, the authors consider instead a simpler class of systems.
- The justification for changing the class of systems is simple-most of the systems encountered as examples do not have the complicated structure of their canonical example.
- Thus there may be a simpler class of systems which is both steerable using simple sinusoids and representative of systems in which the authors are interested.

### is, in

- The same approach can be used to steer x , through x5.the authors.
- The first item is to check the controllability of these systems.
- To steer this system, the authors use sinusoids at integrally related frequencies.
- Direct computation starting from the origin yields.
- Let hfj represent the motion corresponding to the Lie product Xi.

### C. Noncanonical Chained Systems

- Hence, the authors cannot use simple sinusoids to steer this system as before.
- Since the automobile had degree of nonholonomy 3, the problems present in the previous example do not occur.
- Similar efforts have been used by Lafferriere and Sussmann [261 to convert systems into nilpotent form for use with their planning algorithm.
- Finally, sinusoids may be useful for steering systems.
- The following example illustrates the limitations of using sinusoidal which ake not locally in canonical form.

### D. Converting Systems to Chained Form

- This set of conditions gives a constructive method for building a feedback transformation which accomplishes the conversion.
- The motion is implemented as a feedback precompensator which converts the U inputs into the actual system input, U. This feedback transformation agrees the that used in Lafferriere and Sussmann to nilpotentize the kinematic car example.
- The results of Section I11 are somewhat complimentary-the methods can easily be applied to certain systems which are not nilpotent, but the general case requires a restrictive canonical form.
- The work in nonholonomic motion planning thus far has been primarily in the generation of open-loop trajectories.
- Recent work by Coron has shown that it is possible to stabilize a nonholonomic system using smooth, time-varying state feedback McClamroch have studied problems related to stabilization to a manifold instead of a point [31, [41.

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##### Citations

6,592 citations

6,340 citations

### Cites background or methods from "Nonholonomic motion planning: steer..."

...The model given here is adapted from [730]....

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...The steering method presented in this section is based on initial work by Brockett [144] and a substantial generalization of it by Murray and Sastry [730]....

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...The presentation given here is based on [730, 848]....

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...For more details beyond the presentation here, see [599, 728, 730, 848]....

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...For a proof of the correctness of the second phase, and more information in general, see [730, 848]....

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3,025 citations

### Cites background from "Nonholonomic motion planning: steer..."

...Novel and Lévine 1990, Marttinen et al. 1990) and the car with n trailers (Murray and Sastry 1993, Rouchon et al. 1993a)....

[...]

...This implies the existence of non-flat systems which verify the strong accessibility property (Sussmann and Jurdjevic 1972)....

[...]

...The motion planning of the car with n-trailer is perhaps the most popular example of path planning of nonholonomic systems (Laumond 1991, Murray and Sastry 1993, Monaco and Normand-Cyrot 1992, Rouchon et al. 1993a, Tilbury et al. 1993, Martin and Rouchon 1993, Rouchon et al. 1993b)....

[...]

...The resulting dynamics are described by the following equations (the notations are those of (Murray and Sastry 1993) and summarized on figure 3): ẋ0 = u1 cos θ0 ẏ0 = u1 sin θ0 φ̇ = u2 θ̇0 = u1 d0 tan φ θ̇i = u1 di ( i−1∏ j=1 cos(θj−1 − θj) ) sin(θi−1 − θi) for i = 1, . . . , n (18) where (x0,…...

[...]

...Steering a car with n trailers is now the object of active researches (Laumond 1991, Murray and Sastry 1993, Monaco and Normand-Cyrot 1992, Rouchon et al. 1993a, Tilbury et al. 1993)....

[...]

2,993 citations

### Cites background from "Nonholonomic motion planning: steer..."

...Techniques also exist for general system classes, such as nilpotent (Laffierriere and Sussman 1991), differentially flat (Murray, Rathinam, and Sluis 1995; Fliess et al. 1993), and chained form (Bushnell, Tilbury, and Sastry 1995; Murray and Sastry 1993; Struemper 1997)....

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...For more complicated kinematic models, nonoptimal steering techniques have been introduced, which include for example a car pulling trailers (Murray and Sastry 1993) and fire trucks (Bushnell, Tilbury, and Sastry 1995)....

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2,954 citations

### Cites background from "Nonholonomic motion planning: steer..."

...Output-feedback control simulation Consider a simple kinematic model of car control (Murray and Sastry 1993) _x ¼ v cos’; _y ¼ v sin’ _’ ¼ ðv=lÞ tan _ ¼ u where x and y are Cartesian coordinates of the rear-axle middle point, ’ is the orientation angle, v is the longitudinal velocity, l is the…...

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...Consider a simple kinematic model of car control ( Murray and Sastry 1993 )...

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##### References

11,008 citations

8,525 citations

6,186 citations

### "Nonholonomic motion planning: steer..." refers background in this paper

...This has resulted in a rather complete understanding of the complexity of the computational effort required to plan the trajectories of robots to avoid both fixed and moving obstacles [8, 25, 19]....

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3,840 citations

3,063 citations

### "Nonholonomic motion planning: steer..." refers background in this paper

...Other approaches include the use of potential functions for navigating in cluttered envi ronments [22, 21] and compliant motion planning for navigating in the presence of uncertainty [10, 11, 34]....

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