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## A Direct Method for Trajectory Optimization of Rigid Bodies Through Contact (2012)

Citations: | 7 - 2 self |

### Citations

571 | SNOPT: An SQP algorithm for large-scale constrained optimization
- GILL, MURRAY, et al.
- 2002
(Show Context)
Citation Context ...nerally helpful in avoiding problems with local minima. The resulting optimization problems are also sparse, allowing efficient (locally optimal) solutions with largescale sparse solvers such as SNOPT=-=[9]-=-, and trivial parallel/distributed evaluation of the cost and constraints. In this paper, we consider the problem of designing direct trajectory optimization methods for rigid-body systems with contac... |

240 |
1998] Survey of numerical methods for trajectory optimization
- Betts
(Show Context)
Citation Context ...ll under the umbrella of multiple shooting. There are advantages and disadvantages to both direct and shooting methods, which are outlined here. For a more detailed comparison, see the survey article =-=[6]-=-. For many problems, direct methods enjoy a considerable numerical advantage over the shooting methods, which can be plagued by poorly conditioned gradients; for instance, a small change in the contro... |

234 |
Practical Methods for Optimal Control Using Non-Linear Programming
- Betts
- 2001
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Citation Context ... ∗ A preliminary version of this paper was presented at the 2012 Workshop on the Algorithmic Foundations of Robotics (WAFR) conference[14] 1finitely parameterized nonlinear optimization problem (see =-=[3]-=-). Broadly speaking, these transcriptions fall into two categories: the shooting methods and the direct methods. In shooting methods, the nonlinear optimization searches over (a finite parameterizatio... |

232 |
Efficient bipedal robots based on passive–dynamic walkers
- Collins, Ruina, et al.
- 2005
(Show Context)
Citation Context ...o generate a cyclic walking gait and the trajectory was optimized for mechanical cost of transport. Cost of transport is a common, unitless indication of the energy consumption required for locomotion=-=[6]-=-, and the “mechanical” cost of transport is computed using the total positive work done on the system independent of losses in the actuators or costs due to onboard electronics. Where d is the total d... |

193 | An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and Coulomb friction
- Stewart, Trinkle
- 1996
(Show Context)
Citation Context ...tiple passive springs and tendons. The legs are driven at the hip to keep the leg mass as low as possible. plementarity problem (LCP) which can be solved with well known methods like Lemke’s Algorithm=-=[21]-=-. Significant additional work has been done to extend the original LCP methods to guarantee solutions to multi-body contact[2, 12]. In many cases, solutions to the LCP for forward simulation can be fo... |

150 | Design and Use Paradigms for Gazebo, An OpenSource Multi-Robot Simulator
- Koenig, Howard
- 2004
(Show Context)
Citation Context ...robotic manipulation which plan through contact - most planners plan up to a pre-grasp then activate a separate grasping controller. Indeed, the multicontact dynamics engines used to simulate grasping=-=[11, 12]-=- do not use hybrid models of the dynamics, because the permutations of different possible modes grows exponentially with the number of links and contact points, and because hybrid models can be plague... |

144 |
Differential dynamic programming
- Jacobson, Mayne
- 1970
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Citation Context ...on problem (see [7]). Broadly speaking, these transcriptions fall into two categories: the shooting methods and the direct methods. In shooting methods, such as Differential Dynamic Programming (DDP) =-=[20]-=-, the nonlinear optimization searches over (a finite parameterization of) u(t), using a forward simulation from x(0) to evaluate the cost of every candidate input trajectory. In direct methods, the no... |

136 |
Direct Trajectory Optimization Using Nonlinear Programming and Collocation
- Hargraves, Paris
- 1987
(Show Context)
Citation Context ...r parameterizations of u(t) and x(t); here no simulation is required and instead the dynamics are imposed as a set of optimization constraints, typically evaluated at a selection of collocation points=-=[10]-=-. Mixtures of shooting and direct methods are also possible, and fall under the umbrella of multiple shooting. Direct methods typically enjoy a considerable numerical advantage over the shooting metho... |

134 |
A special Newton-type optimization method
- Fischer
- 1992
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Citation Context ...e to a better trajectory by improving the conditioning of the SQP. Many smoothing functions for NCPs exist and have been used to directly solve these problems, such as the Fischer-Burmeister function =-=[8]-=- or the class of functions suggested in [5], and these functions may be applicable here as well. As is mentioned above, throughout this paper we have solely dealt with inelastic collisions where the e... |

131 |
A class of smoothing functions for nonlinear and mixed complementarity problems
- Chen, Mangasarian
- 1996
(Show Context)
Citation Context ...onditioning of the SQP. Many smoothing functions for NCPs exist and have been used to directly solve these problems, such as the Fischer-Burmeister function [8] or the class of functions suggested in =-=[5]-=-, and these functions may be applicable here as well. As is mentioned above, throughout this paper we have solely dealt with inelastic collisions where the effective coefficient of restitution vanishe... |

130 | Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. Nonlinear Dynamics
- Anitescu, Potra
- 1997
(Show Context)
Citation Context ... (LCP) which can be solved with well known methods like Lemke’s Algorithm[21]. Significant additional work has been done to extend the original LCP methods to guarantee solutions to multi-body contact=-=[2, 12]-=-. In many cases, solutions to the LCP for forward simulation can be found via the minimization of a convex quadratic program (QP). In this paper we demonstrate that it is possible, indeed natural, to ... |

130 | Feedback Control of Dynamic Bipedal Robot Locomotion
- Westervelt, Grizzle, et al.
- 2007
(Show Context)
Citation Context ...g trajectory optimization very inefficient. For numerical efficiency, a preferred method is to approximate contact as an autonomous hybrid dynamical system that undergoes discontinuous switching (see =-=[25]-=-). The discrete transitions are fully autonomous as we can directly control neither the switching times nor the switching surface. There are a number of impressive success stories for trajectory optim... |

128 |
Rigid-body dynamics with friction and impact
- Stewart
(Show Context)
Citation Context ...lision). In the limiting case of purely rigid bodies, the constraint forces, λ(t), are often modeled with the Dirac δ functions, as hybrid impulsive events, or through measure differential inclusions =-=[20, 1]-=-. However, many of these complexities can be avoided by discretizing the system in time. [21] introduced a time-stepping method that only considers the integral of contact forces over a period and so ... |

86 | On dynamic multi-rigid-body contact problems with coulomb friction. Zeitschrift für Angewandte Mathematik und Mechanik
- Trinkle, Pang, et al.
- 1997
(Show Context)
Citation Context ...blem In order to avoid the combinatorial explosion of hybrid models, simulation techniques in computer graphics and in grasping research use a different formulation of contact, summarized briefly here=-=[23]-=-. The forward dynamics of a rigid-body (e.g., with a floating base) subject to frictionless contact constraints can be written as find ¨q,λ (1) subject to H(q) ¨q +C(q, ˙q) + G(q) = B(q)u + J(q) T λ, ... |

82 |
Numerical Methods for Nonsmooth Dynamical Systems
- Acary, Brogliato
- 2008
(Show Context)
Citation Context ... straightforward approach to extending to 3D would be to treat both λx and ψ(q, q̇) as two-vectors, and write down a set of nonlinear constraints for Coulomb friction in the tangent plane, such as in =-=[1]-=-. However, to preserve the MPCC structure of our problem, we instead use a polyhedral approximation of the friction cone, as in [34]. Let Di for i = 1, ..,d be unit vectors in R2 whose convex hull is ... |

74 | Intuitive Control of a Planar Bipedal Walking Robot
- Pratt, Pratt
- 1998
(Show Context)
Citation Context ...rohibit these forces from crossing a specified threshold. 3.3 Spring Flamingo Walking Gait To analyze a more realistic system, we tested our methods on a planar simulation of the Spring Flamingo robot=-=[16]-=-. On Spring Flamingo, each leg has three actuated joints (hip, knee, and ankle) and there are contact points at the toe and heel of each foot. Many hybrid walking models use a constrained form of the ... |

70 | Local convergence of SQP methods for mathematical programs with equilibrium constraints
- Fletcher, Leyffer, et al.
(Show Context)
Citation Context ...een done in this area, and we leverage Sequential Quadratic Programming (SQP) techniques - a particular class of algorithms for solving general nonlinear programs that have been shown to be effective =-=[3, 16]-=-. Broadly speaking, SQP solves a sequence of quadratic programs which each approximate the original nonlinear program. The key to this formulation is in resolving the contact forces, the mode-dependen... |

68 | Computer optimization of a minimal biped model discovers walking and running
- Srinivasan, Ruina
(Show Context)
Citation Context ...y difficult to remove this assumption in the direct methods. Slight variations from the original sequence are possible if the formulation allows the time duration of individual modes to vanish, as in =-=[19]-=-. For problems with fewer possi4ble modes, outer optimization loops have been used to determine the hybrid mode schedule, as in [24]. In some cases, the combinatorial problem of solving for a mode sc... |

66 | Exploiting Inherent Robustness and Natural Dynamics in the Control of Bipedal Walking Robots
- Pratt
- 2000
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Citation Context ...,l , (39) xN,CM ≥ x1,CM +dmin. (40) With these linear constraints and given a nominal trajectory from Pratt’s original work on the robot where the mechanical cost of transport was 0.18 =-=[28]-=-, our 15 Figure 7: A generated trajectory for the FastRunner robot running at over 20 mph. The solid elements show the leg linkages and the thin lines indicate springs and tendons. Only the hip joints... |

57 |
Nonsmooth Mechanics: Models, Dynamics and Control
- Brogliato
- 1999
(Show Context)
Citation Context ...netrating configuration space is non-convex, although work has been done in this area regarding simulation [26]. Standard variations on this formulation can be made to also address frictional contact =-=[37, 8]-=-. The solutions to these dynamics are potentially complex, involving large impact forces occurring over very short time periods (e.g., at the moment of a new collision). In the limiting case of purely... |

55 |
A globally convergent sequential quadratic programming algorithm for mathematical programming with linear complementarity constraints
- Fukushima, ZQ, et al.
- 1998
(Show Context)
Citation Context ...es, we can consider equivalent formulations of these complementarity conditions. Fukushima et al. propose an iterative method that sequentially tightens relaxations of the complementarity constraints =-=[17]-=-. In our work, we primarily adopt the scheme of Anitescu who proposed leveraging the elastic mode of SQP solvers like SNOPT to solve a set of similar, and equivalent constraints [3], G(x)≥ 0 (23) H(x)... |

41 | Optimal gait and form for animal locomotion
- WAMPLER, POPOVIĆ
- 2009
(Show Context)
Citation Context ...tion allows the time duration of individual modes to vanish, as in [19]. For problems with fewer possi4ble modes, outer optimization loops have been used to determine the hybrid mode schedule, as in =-=[24]-=-. In some cases, the combinatorial problem of solving for a mode schedule have been addressed by combinatorial planners; a variant of the Rapidly-Exploring Random Tree (RRT) algorithm was used in [18]... |

37 | Synthesis and stabilization of complex behaviors through online trajectory optimization. IROS
- Tassa, Erez, et al.
- 2012
(Show Context)
Citation Context ...with a form of Model Predictive Control (MPC). Robust, contact invariant MPC of underactuated systems remains an open problem, although there has been significant recent work in this area, such as in =-=[35]-=-. We believe that, given a nominal trajectory, this approach can be adapted to solve the problem of real-time local control and that the controller will be capable of planning a finite-horizon path wi... |

30 | Approximate optimal control of the compass gait on rough terrain
- Byl, Tedrake
(Show Context)
Citation Context ...e bounding trajectories for a quadruped over terrain. Methods for optimal control which approximate the global optimal, such as brute force methods based on dynamic programming, have also been applied=-=[4]-=-, but are so far limited to low dimensional problems. 1.2 Contact Dynamics as a Complementarity Problem In order to avoid the combinatorial explosion of hybrid models, simulation techniques in compute... |

30 | 2003. Implementation of multi-rigid-body dynamics within a robotic grasping simulator
- MILLER, CHRISTENSEN
(Show Context)
Citation Context ...robotic manipulation which plan through contact - most planners plan up to a pre-grasp then activate a separate grasping controller. Indeed, the multicontact dynamics engines used to simulate grasping=-=[11, 12]-=- do not use hybrid models of the dynamics, because the permutations of different possible modes grows exponentially with the number of links and contact points, and because hybrid models can be plague... |

22 | Discovery of complex behaviors through contact-invariant optimization
- Mordatch, Todorov, et al.
- 2012
(Show Context)
Citation Context ...where the simulated dynamics of the initial tape do not find the desired mode sequence, no gradient information will be available to guide the local search. In another recent approach, Mordatch et al.=-=[13]-=- have also explored relaxations of the the complementarity contact constraints in the context of trajectory optimization. These methods have been remarkably successful at generating natural looking mo... |

22 | Direct trajectory optimization of rigid body dynamical systems through contact
- Posa, Tedrake
- 2012
(Show Context)
Citation Context ...ethods for transcribing the trajectory optimization problem into a ∗ A preliminary version of this paper was presented at the 2012 Workshop on the Algorithmic Foundations of Robotics (WAFR) conference=-=[14]-=- 1finitely parameterized nonlinear optimization problem (see [3]). Broadly speaking, these transcriptions fall into two categories: the shooting methods and the direct methods. In shooting methods, t... |

22 | Bounding on rough terrain with the LittleDog robot
- Shkolnik, Levashov, et al.
- 2011
(Show Context)
Citation Context ...[24]. In some cases, the combinatorial problem of solving for a mode schedule have been addressed by combinatorial planners; a variant of the Rapidly-Exploring Random Tree (RRT) algorithm was used in =-=[18]-=- to produce bounding trajectories for a quadruped over terrain. Methods for optimal control which approximate the global optimal, such as brute force methods based on dynamic programming, have also be... |

20 |
On the equivalence between complementarity systems, projected systems and differential inclusions
- Brogliato, Daniilidis, et al.
(Show Context)
Citation Context ...lision). In the limiting case of purely rigid bodies, the constraint forces, λ(t), are often modeled with the Dirac δ functions, as hybrid impulsive events, or through measure differential inclusions =-=[20, 1]-=-. However, many of these complexities can be avoided by discretizing the system in time. [21] introduced a time-stepping method that only considers the integral of contact forces over a period and so ... |

20 |
On using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints
- Anitescu
- 2005
(Show Context)
Citation Context ...een done in this area, and we leverage Sequential Quadratic Programming (SQP) techniques - a particular class of algorithms for solving general nonlinear programs that have been shown to be effective =-=[3, 16]-=-. Broadly speaking, SQP solves a sequence of quadratic programs which each approximate the original nonlinear program. The key to this formulation is in resolving the contact forces, the mode-dependen... |

17 | Stochastic complementarity for local control of discontinuous dynamics. Robotics: Science and Systems
- Tassa, Todorov
- 2010
(Show Context)
Citation Context ...ands of simultaneous contacts. Despite the success in forward simulation, an implicit complementarity based formulation has only recently been used in direct trajectory optimization. In related work, =-=[22]-=- explored the use of Stochastic Complementarity Problems in a shooting method. Their approach does not require an explicit mode schedule, but instead relies on the passive dynamics of the system to in... |

12 |
Contact-invariant optimization for hand manipulation
- Mordatch, Popović, et al.
- 2012
(Show Context)
Citation Context ...tion we do not make in the current paper. The work of Erez and Todorov 6 optimized a human running gait by smoothing the contact dynamics to make use of a custom inverse dynamics formulation [14]. In =-=[24]-=-, Mordatch et al. include the contact forces as optimization parameters, similar to the work in this paper and our preliminary version [27]. Mordatch includes the feasibility of the contact forces as ... |

11 |
Modeling and Optimal Control of Human-Like Running
- Schultz, Mombaur
- 2010
(Show Context)
Citation Context ...ing times nor the switching surface. There are a number of impressive success stories for trajectory optimization in these hybrid models, for instance the optimization of an impressive 3D running gait=-=[17]-=-. These results primarily use direct methods. But they are plagued with one major short-coming - the optimization is constrained to operate within an a priori specification of the ordering of hybrid m... |

10 |
Trajectory optimization for domains with contacts using inverse dynamics. IROS
- Erez, Todorov
- 2012
(Show Context)
Citation Context ...an assumption we do not make in the current paper. The work of Erez and Todorov 6 optimized a human running gait by smoothing the contact dynamics to make use of a custom inverse dynamics formulation =-=[14]-=-. In [24], Mordatch et al. include the contact forces as optimization parameters, similar to the work in this paper and our preliminary version [27]. Mordatch includes the feasibility of the contact f... |

6 |
Fastrunner: A fast, efficient and robust bipedal robot. concept and planar simulation
- Cotton, Olaru, et al.
- 2012
(Show Context)
Citation Context ...st way to solve complex trajectories without the need for a mode schedule. Specifically, this work was motivated by the challenge of optimizing trajectories for a new running robot called “FastRunner”=-=[7]-=-. FastRunner, illustrated in Figure 1 is a bipedal robot concept designed to run at speeds over 20 mph and up to 50 mph. Most notably, FastRunner has an exceptionally clever, but also exceptionally co... |

6 | A sparse model predictive control formulation for walking motion generation
- Dimitrov, Sherikov, et al.
- 2011
(Show Context)
Citation Context ...parse solvers such as SNOPT [18], and trivial parallel/distributed evaluation of the cost and constraints. Dimitrov et al. demonstrate the numerical benefits of sparse, direct optimization for MPC in =-=[13]-=-. We also note that throughout the optimization process, shooting methods that determine the state trajectory through simulation will always result in dynamically feasible trajectories. However, in th... |

4 |
Modeling non-convex configuration space using linear complementarity problems
- NGUYEN, TRINKLE
- 2010
(Show Context)
Citation Context ...t. It should be noted that (3) may be difficult to pose properly in the case where the non-penetrating configuration space is non-convex, although work has been done in this area regarding simulation =-=[26]-=-. Standard variations on this formulation can be made to also address frictional contact [37, 8]. The solutions to these dynamics are potentially complex, involving large impact forces occurring over ... |

2 | Sources of error in a simulation of rigid parts on a vibrating rigid plate
- Berard, Nguyen, et al.
(Show Context)
Citation Context ... of simultaneous contacts. 2.3 Related Work A number of researchers are currently pursuing mode invariant trajectory optimization, as contrasted with the traditional hybrid systems based approach. In =-=[5]-=-, Berard et al. used the LCP formulation of contact to design trajectories of single body on a vibrating plate. Here, they optimize over a small set of parameters describing the oscillating behavior o... |

2 |
J.S.Pang and D.Ralph, ”Mathematical Programs with Equilibrium Constraints
- Luo
- 1996
(Show Context)
Citation Context ...nonlinear optimization for trajectory design, resulting in a Mathematical Program with Complementarity Constraints (MPCC), or, equivalently, a Mathematical Program with Equilibrium Constraints (MPEC) =-=[22]-=-. While these are generally difficult to solve, significant research has been done in this area, and we leverage Sequential Quadratic Programming (SQP) techniques - a particular class of algorithms fo... |