Optimal Control of a Rigid Body using Geometrically Exact Computations on SE(3)

Melvin Leok; N. Harris McClamroch; Taeyoung Lee

REVIEW

Optimal Control of a Rigid Body using Geometrically Exact Computations on SE(3)

Taeyoung Lee, N. Harris McClamroch, Melvin Leok

Optimization and Control · Submitted 26 Feb 2006 · 13 references

Pith's verdict: A clean SE(3) optimal control framework whose central sensitivity matrix is omitted and then mismatched to the wrong integrator; the core idea is sound but the paper is not self-contained.

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CitationMelvin Leok, N. Harris McClamroch, Taeyoung Lee. "Optimal Control of a Rigid Body using Geometrically Exact Computations on SE(3)." arXiv:math/0602588 (2006). https://pith.science/paper/INPRQ2K4

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Claim paper

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Editor's note

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The thing to know: this is a genuine extension of the authors' SO(3) work to full rigid-body motion on SE(3), including dynamics and potential forces, via a Lie group variational integrator. The formulation is clean, the examples are internally consistent, and the convergence plots look like Newton iteration on their discrete problem. That is real value.

What's actually new: the paper casts impulsive and smooth optimal control as, respectively, a parameter optimization and a shooting problem on SE(3), with discrete necessary conditions obtained by constrained variations. That framework is a reasonable contribution and should be citable as an early reference in geometric optimal control.

The soft spots, in order of severity. First, the central object A_k is never given. In Section III it is simply declared "can be suitably defined," and in Section IV the sensitivity calculations explicitly reference (18). But Section IV solves a different, modified first-order system (21)-(25) that omits the h^2 term in (7), uses one-sided force/moment terms instead of the midpoint terms in (8) and (11), and replaces the shifted S argument in (9) with S(Π_k). The stress-test note is right: the adjoint equation (28) and the shooting sensitivity Ψ12 are therefore not derived for the system actually solved. This is a real inconsistency, not a cosmetic gap, and it directly undermines the claim that the gradients are exact.

Second, there is no comparison against a non-geometric baseline—say, quaternions with normalization or a direct transcription method—so the claimed advantage of the geometric approach is demonstrated only by internal consistency. That is weaker than the authors imply.

Third, the self-citation load is heavy but not illegitimate: [6] is the source of the integrator, so depending on it is fair. Still, the paper would be much stronger if it were self-contained about the linearization.

On the whole, the central idea is defensible and probably correct; the paper is incomplete and internally mismatched in Section IV, but not incoherent. The machine-precision convergence is convergence to the discrete optimum, not to the continuous solution, so "geometrically exact" should be read carefully.

Who this is for: people working in geometric mechanics, spacecraft trajectory optimization, or computational optimal control. It deserves a serious referee, but the referee should require the explicit A_k for both integrators, a re-derivation of the Section IV sensitivities against the modified equations, and a baseline comparison. I would accept it for peer review, not for publication as is.

Referee report

Significance

If the derivations were completed and verified, the paper would offer a useful contribution: it handles rigid-body optimal control globally on SE(3), avoids local charts and quaternion ambiguities, and reduces the shooting iteration to a small number of unknown initial multipliers. The numerical examples suggest that the approach can drive terminal constraint errors to machine precision. However, the central correctness issues are currently unresolved: the key linearization matrix A_k is never given explicitly, and the sensitivity analysis for the smooth-control problem is based on the linearization of a different integrator than the one actually used in the necessary conditions. These gaps prevent verification of the claimed exact sensitivities and of the numerical convergence properties.

Major objections (3)

[Section III-B, Eq. (18)] The central sensitivity matrix A_k is never specified. Equation (18) states that A_k 'can be suitably defined', and the same unspecified object is reused in Equations (28)-(31) of Section IV-C. Since the claimed exact gradients, the shooting direction D = Ψ_{12}^{-1}, and the Newton-Armijo convergence all depend on this matrix, a reader cannot verify or reproduce any of the numerical results. The authors must provide the explicit entries of A_k and either derive them from the linearized equations or supply an independent validation (e.g., a finite-difference check) of the resulting sensitivities.
[Section IV-A, IV-B, IV-C, Eqs. (21)-(31)] The necessary conditions (21)-(25) are stated for a modified first-order integrator that differs from the original integrator (7)-(11): Equation (21) drops the h^2 acceleration term of (7); Equations (22) and (25) use h f_{k+1} and h M_{k+1} instead of the midpoint terms in (8) and (11); and Equation (23) uses h S(Π_k) rather than h S(Π_k + (h/2)(M_k + u_k^m)) from (9). Nevertheless, the adjoint equation (28) and the sensitivity equations (29)-(31) reuse the matrix A_k from (18), which is the linearization of the original integrator (7)-(11). The linearized map for the modified equations (21)-(25) has different entries, so the derived sensitivities do not correspond to the system actually solved in the numerical examples. The authors must either derive A_k for the modified equations or prove that the two linearizations coincide.
[Section IV-B, augmented performance index] The augmented performance index contains the term λ3,k^T S^{-1}( logm(F_k - R_k^T R_{k+1}) ). Under the update (10) (and its modified counterpart (24)), R_{k+1} = R_k F_k, so F_k - R_k^T R_{k+1} is the zero matrix and logm(0) is undefined. This term, and the constrained-variation calculation built on it, are not well-posed as written. The authors should correct the expression, presumably to involve logm(R_k^T R_{k+1} F_k^{-1}) or another well-defined quantity, and re-derive the resulting necessary conditions.

Minor objections (6)

[Section IV-A] The first sentence says 'An optimal impulsive control problem is formulated', but the performance index and the control variables describe smooth controls applied at every time step. This appears to be a typo; it should read 'optimal control problem'.
[Section III-B, Eq. (12)] The notation R_k S(ζ_k) is used for the variation of the attitude, which is clear, but the paper never defines the relationship between ζ_k and the exponential-map coefficient η_k used in the preceding line. A brief clarification would help the reader follow the constrained variation derivation.
[Section V, Newton-Armijo algorithm] The stopping criterion ǫ_S and the scaling factor α in the Newton-Armijo iteration are never given numerical values, and the number of line-search iterations per outer step is not reported. This makes the reported iteration counts hard to reproduce.
[Section V, convergence claims] The text claims that the convergence results are 'consistent with the quadratic convergence rates expected of Newton methods', but no slope or rate is measured in the figures, and no comparison with another solver or with a finite-difference sensitivity computation is provided. A quantitative convergence-rate statement would strengthen the claim.
[Section V, neutral stability discussion] The paragraph asserting that the discrete multiplier equations 'preserve the neutral stability property numerically' is not supported by any analysis or computation. This is a strong claim about the numerical method and should either be proved or explicitly labeled as an empirical observation.
[References] The paper relies on the self-cited submitted manuscripts [6] and [9] for the integrator and for the SO(3) counterpart. If published versions are available, they should be cited instead; otherwise the dependence on unpublished work should be noted in the text.

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The paper

The paper claims

Rigid-body optimal control solved to machine precision on SE(3)

The reading

This paper tries to establish that optimal control problems for a rigid body moving and rotating in three dimensions can be formulated and solved directly on the Lie group SE(3), the space of positions and attitudes, without local charts or coordinate singularities. The authors discretize the dynamics with a Lie group variational integrator, derive discrete optimality conditions in the Lie algebra, and compute sensitivity derivatives from a linearized variational equation. They report that this yields accurate gradients, Newton-Armijo shooting iteration that converges quadratically, and terminal boundary errors near machine precision (on the order of $10^{-13}$ to $10^{-15}$) for spacecraft orbital-and-attitude maneuvers such as orbital radius change, inclination change, and orbital capture. If correct, the paper shows that geometric structure preservation is not just a conservation nicety but a practical tool for fast, accurate optimal control of spacecraft and other rigid bodies.

What carries the argument

The central object is the Lie group variational integrator on SE(3), defined by equations (7)--(11): a trapezoidal translation update, an attitude update $R_{k+1} = R_k F_k$ where $F_k \in SO(3)$ solves $hS(\Pi_k + \frac{h}{2}(M_k + u^m_k)) = F_k J_d - J_d F_k^T$, and momentum updates in the body frame. All infinitesimal variations are expressed in the Lie algebra $\mathfrak{se}(3)$ using homogeneous coordinates and the exponential map, so the linearized map $z_{k+1} = A_k z_k$ and its terminal sensitivity $\Phi$, and the analogous $\Psi_{12}$ for the shooting method, are computed without coordinates. These sensitivities are then used to supply gradients for an SQP method in the impulsive-control case and a Newton-Armijo neighboring-extremal iteration in the smooth-control case.

Extended reading notes

Core claim

The central claim, stated in Section I and supported by the numerical examples, is that geometrically exact computations on SE(3) provide an efficient and accurate way to solve optimal control problems of a rigid body whose translational and rotational dynamics are coupled through a potential. The paper shows that replacing the usual Lie group kinematics equation with a discrete update $R_{k+1} = R_k F_k$, where $F_k$ solves an implicit equation in the Lie algebra, and expressing all variations in the Lie algebra via the exponential map, yields discrete necessary conditions (21)--(28) whose sensitivity matrices are accurate enough for a neighboring-extremal shooting method to drive terminal constraint violations to machine precision. It further claims that this avoids the singularities of Euler angles and the drift or double-cover ambiguity of quaternions, and that the integrator's preservation of symplectic, momentum, and neutral-stability properties is what makes the computed gradients reliable.

Load-bearing premise

The load-bearing premise is that the linearization matrix $A_k$, whose entries the paper never writes down, correctly represents the discrete variational equations, so that the sensitivity matrix computed from it exactly relates terminal errors to initial multiplier perturbations; if that linearization is wrong, the claimed exact gradients and quadratic convergence do not follow.

Editorial extensions

If this is right

Combined orbital and attitude maneuvers can be computed with terminal boundary errors near machine precision, as in the reported values $4.77 \times 10^{-15}$, $3.88 \times 10^{-15}$, $3.35 \times 10^{-13}$, and $3.26 \times 10^{-13}$.
Because the configuration is a global SE(3) element, the method avoids Euler-angle singularities and quaternion drift or ambiguity, so it can handle large-angle rotations without changing coordinates.
The shooting formulation keeps the number of iteration variables at the dimension of the initial multiplier, independent of the number of time steps, unlike methods that optimize the entire control history.
The observed quadratic convergence is consistent with Newton iteration using accurately computed sensitivity derivatives, implying that the linearization captures the discrete variational structure.
Fixed-terminal-state problems become two-point boundary value problems on SE(3), which the paper identifies as an extension of the Hohmann transfer to include attitude changes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the variational and exponential-map perturbation structure is group-theoretic rather than specific to SE(3), the same recipe of discretizing with a Lie group variational integrator, linearizing in the Lie algebra, and shooting on the initial multiplier should extend to other configuration Lie groups, such as SE(n) or products of SO(3) and $\mathbb{R}^n$ for multibody systems.
A direct check of the accuracy claim is available: the matrix $A_k$ in equation (18) is never given, so computing it explicitly (or replacing it with a finite-difference Jacobian of the integrator map) and re-running the shooting iteration would confirm whether the quadratic convergence is attributable to that linearization or to the integrator's geometry preservation.
The same sensitivity machinery could be embedded in model-predictive control or direct transcription with geometric integrators, where each receding-horizon solve would inherit the accurate gradients and fast Newton convergence demonstrated here.

Original abstract

Optimal control problems are formulated and efficient computational procedures are proposed for combined orbital and rotational maneuvers of a rigid body in three dimensions. The rigid body is assumed to act under the influence of forces and moments that arise from a potential and from control forces and moments. The key features of this paper are its use of computational procedures that are guaranteed to preserve the geometry of the optimal solutions. The theoretical basis for the computational procedures is summarized, and examples of optimal spacecraft maneuvers are presented.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,535 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.07 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Optimal control problems are formulated and efficient computational procedures are proposed for combined orbital and rotational maneuvers of a rigid body in three dimensions. The rigid body is assumed to act under the influence of forces and moments that arise from a potential and from control forces and moments. The key features of this paper are its use of computational procedures that are guaranteed to preserve the geometry of the optimal solutions. The theoretical basis for the computational procedures is summarized, and examples of optimal spacecraft maneuvers are presented.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Optimal control problems are formulated and efficient computational procedures are proposed for combined orbital and rotational maneuvers of a rigid body in three dimensions.	contribution 0.99	0.88	0.25	asserted 1.00
s1	The rigid body is assumed to act under the influence of forces and moments that arise from a potential and from control forces and moments.	background 0.54	0.23	0.04	asserted 0.99
s2	The key features of this paper are its use of computational procedures that are guaranteed to preserve the geometry of the optimal solutions.	contribution 0.99	0.87	0.44	asserted 1.00
s3	The theoretical basis for the computational procedures is summarized, and examples of optimal spacecraft maneuvers are presented.	contribution 0.89	0.83	0.13	asserted 0.98

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.90.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** TPBVP: Orbital radius change REFERENCES [1] K. Spindler, “Optimal control on Lie groups with applications to attitude control,” Mathematics of Control, Signals, and Systems, vol. 11, pp. 197–219, 1998. [2] S. Sastry, “Optimal control on Lie groups,” in Proceedings of the Third International Congress on Industrial and Applied Mathematics (ICIAM), 1995. [3] T. Lee, M. Leok, and N. H. McClamroch, “A Lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum,” in Proceedings of the IEEE Conference on Control Applications, 2005, pp.…

**Figure 3.** Optimal control: Orbital inclination change

**Figure 4.** Optimal control: Orbital capture

Showing 3 of the paper's 4 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:math/0602588 by the authors.

References

Works cited (13)

[6]

T. Lee, M. Leok, and N. H. McClamroch, “Lie group variatio nal integrators for the full body problem,” Computer Methods in Applied Mechanics and Engineering , 2005, submitted. [Online]. Available: http://arxiv.org/math.NA/0508365

View on Pith arXiv 2005
[1]

K. Spindler, “Optimal control on Lie groups with applica tions to atti- tude control,” Mathematics of Control, Signals, and Systems , vol. 11, pp. 197–219, 1998

View on Pith 1998
[2]

S. Sastry, “Optimal control on Lie groups,” in Proceedings of the Third International Congress on Industrial and Applied Mat hematics (ICIAM), 1995

View on Pith 1995
[3]

T. Lee, M. Leok, and N. H. McClamroch, “A Lie group variati onal integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum,” in Proceedings of the IEEE Conference on Control Applications, 2005, pp. 962–967

View on Pith 2005
[4]

J. E. Marsden and M. West, “Discrete mechanics and variat ional integrators,” Acta Numerica, vol. 10, pp. 357–514, 2001

View on Pith 2001
[5]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, and A. Zanna , “Lie-group methods,” Acta Numerica, vol. 9, pp. 215–365, 2000

View on Pith 2000
[7]

——, “Attitude maneuvers of a rigid spacecraft in a circul ar orbit,” in Proceedings of the American Control Conference , 2006, pp. 1742–1747. [Online]. Available: http://arxiv.org/math.NA/0509299

View on Pith arXiv 2006
[8]

R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation . CRC Press, 1993

View on Pith 1993
[9]

T. Lee, M. Leok, and N. H. McClamroch, “Optimal attitude c ontrol of a rigid body using geometrically exact computations on SO (3),” Journal of Optimization Theory and Applications , 2006, submitted. [Online]. Available: http://arxiv.org/math.OC/0601424

View on Pith arXiv 2006
[10]

A. E. Bryson and Y .-C. Ho, Applied Optimal Control . Hemisphere Publishing Corporation, 1975

View on Pith 1975
[11]

C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations . SIAM, 1995

View on Pith 1995
[12]

J. M. A. Danby, Fundamentals of Celestial Mechanics . Willmann Bell Inc., 1988

View on Pith 1988
[13]

J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, 2001. (a) Spacecraft maneuver 0 0.05 0.1 0.15 0.2 0.25Ŧ1 0 1 0 0.05 0.1 0.15 0.2 0.250 1 2 0 0.05 0.1 0.15 0.2 0.25Ŧ5 0 5 x 10 Ŧ3 t (b) Velocity v 0 0.05 0.1 0.15 0.2 0.25Ŧ0.5 0 0.5 0 0.05 0.1 0.15 0.2 0.25Ŧ2 Ŧ1 0 0 0.05 0.1 0.15 0.2 0.25Ŧ1 0 1 t (c) Angular velocity Ω F...

View on Pith 2001

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [6]

T. Lee, M. Leok, and N. H. McClamroch, “Lie group variatio nal integrators for the full body problem,” Computer Methods in Applied Mechanics and Engineering , 2005, submitted. [Online]. Available: http://arxiv.org/math.NA/0508365

View on Pith arXiv 2005

[2] [1]

K. Spindler, “Optimal control on Lie groups with applica tions to atti- tude control,” Mathematics of Control, Signals, and Systems , vol. 11, pp. 197–219, 1998

View on Pith 1998

[3] [2]

S. Sastry, “Optimal control on Lie groups,” in Proceedings of the Third International Congress on Industrial and Applied Mat hematics (ICIAM), 1995

View on Pith 1995

[4] [3]

T. Lee, M. Leok, and N. H. McClamroch, “A Lie group variati onal integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum,” in Proceedings of the IEEE Conference on Control Applications, 2005, pp. 962–967

View on Pith 2005

[5] [4]

J. E. Marsden and M. West, “Discrete mechanics and variat ional integrators,” Acta Numerica, vol. 10, pp. 357–514, 2001

View on Pith 2001

[6] [5]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, and A. Zanna , “Lie-group methods,” Acta Numerica, vol. 9, pp. 215–365, 2000

View on Pith 2000

[7] [7]

——, “Attitude maneuvers of a rigid spacecraft in a circul ar orbit,” in Proceedings of the American Control Conference , 2006, pp. 1742–1747. [Online]. Available: http://arxiv.org/math.NA/0509299

View on Pith arXiv 2006

[8] [8]

R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation . CRC Press, 1993

View on Pith 1993

[9] [9]

T. Lee, M. Leok, and N. H. McClamroch, “Optimal attitude c ontrol of a rigid body using geometrically exact computations on SO (3),” Journal of Optimization Theory and Applications , 2006, submitted. [Online]. Available: http://arxiv.org/math.OC/0601424

View on Pith arXiv 2006

[10] [10]

A. E. Bryson and Y .-C. Ho, Applied Optimal Control . Hemisphere Publishing Corporation, 1975

View on Pith 1975

[11] [11]

C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations . SIAM, 1995

View on Pith 1995

[12] [12]

J. M. A. Danby, Fundamentals of Celestial Mechanics . Willmann Bell Inc., 1988

View on Pith 1988

[13] [13]

J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, 2001. (a) Spacecraft maneuver 0 0.05 0.1 0.15 0.2 0.25Ŧ1 0 1 0 0.05 0.1 0.15 0.2 0.250 1 2 0 0.05 0.1 0.15 0.2 0.25Ŧ5 0 5 x 10 Ŧ3 t (b) Velocity v 0 0.05 0.1 0.15 0.2 0.25Ŧ0.5 0 0.5 0 0.05 0.1 0.15 0.2 0.25Ŧ2 Ŧ1 0 0 0.05 0.1 0.15 0.2 0.25Ŧ1 0 1 t (c) Angular velocity Ω F...

View on Pith 2001