Can we make a robot ballerina perform a pirouette? Orbital stabilization of periodic motions of underactuated mechanical systems

Shiriaev, Anton S.; Freidovich, Leonid B.; Manchester, Ian R.

doi:10.1016/j.arcontrol.2008.07.001

Cited by 97 publications

(87 citation statements)

References 43 publications

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“…Inspired by controlling dynamics in the transversal coordinate [22], we might choose the transversal plane projection, although designing transversal coordinates of the trajectory requires extra effort, and might also be model-specific. Every sample of the cost-to-go matrix is parameterized as decision variables in the optimization program by using the direct collocation approach.…”

Section: Discussionmentioning

confidence: 99%

Optimizing robust limit cycles for legged locomotion on unknown terrain

Dai

Tedrake

2012

2012 IEEE 51st IEEE Conference on Decision and Control (CDC)

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Abstract-While legged animals are adept at traversing rough landscapes, it remains a very challenging task for a legged robot to negotiate unknown terrain. Control systems for legged robots are plagued by dynamic constraints from underactuation, actuator power limits, and frictional ground contact; rather than relying purely on disturbance rejection, considerable advantage can be obtained by planning nominal trajectories which are more easily stabilized. In this paper, we present an approach for designing nominal periodic trajectories for legged robots that maximize a measure of robustness against uncertainty in the geometry of the terrain. We propose a direct collocation method which solves simultaneously for a nominal periodic control input, for many possible one-step solution trajectories (using ground profiles drawn from a distribution over terrain), and for the periodic solution to a jump Riccati equation which provides an expected infinite-horizon cost-to-go for each of these samples. We demonstrate that this trajectory optimization scheme can recover the known deadbeat openloop control solution for the Spring Loaded Inverted Pendulum (SLIP) on unknown terrain. Moreover, we demonstrate that it generalizes to other models like the bipedal compass gait walker, resulting in a dramatic increase in the number of steps taken over moderate terrain when compared against a limit cycle optimized for efficiency only.

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Section: Discussionmentioning

confidence: 99%

Optimizing robust limit cycles for legged locomotion on unknown terrain

Dai

Tedrake

2012

2012 IEEE 51st IEEE Conference on Decision and Control (CDC)

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“…Limit cycles for walking systems in particular are often described as a hybrid dynamics punctuated by discrete impacts. These discrete jump events must be handled with care in the feedback design and verification, but are not fundamentally incompatible with the approach [20]. Figure 4 cartoons the vision of how the algorithm would play out for the well-known compass gait biped [7].…”

Section: Controlling Walking Robotsmentioning

confidence: 99%

LQR-trees: Feedback motion planning on sparse randomized trees

Tedrake¹

2009

Robotics: Science and Systems V

137

134

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Abstract-Recent advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of stability for smooth nonlinear systems. Here we present a feedback motion planning algorithm which uses these results to efficiently combine locally valid linear quadratic regulator (LQR) controllers into a nonlinear feedback policy which probabilistically covers the reachable area of a (bounded) state space with a region of stability, certifying that all initial conditions that are capable of reaching the goal will stabilize to the goal. We investigate the properties of this systematic nonlinear feedback control design algorithm on simple underactuated systems and discuss the potential for control of more complicated control problems like bipedal walking.

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“…In this paper, we use an alternative method based on the combination of transverse linearization and time-varying linear control techniques (Shiriaev et al 2008;Manchester et al 2009;Manchester 2010). This allows one to stabilize more general motions, however a nominal trajectory is required in advance, so this feedback controller must be paired with a motion planning algorithm which takes into account information about the environment.…”

Section: Limit-cycle Approachmentioning

confidence: 99%

“…This is a non-trivial task because of the highly non-linear nature of the dynamics and the limited control authority available for the unactuated coordinate. Several recent efforts in control of underactuated systems have found success in taking a three-stage approach Song and Zefran 2006;Westervelt et al 2007;Shiriaev et al 2008;Manchester et al 2009):…”

Section: Feedback Stabilizationmentioning

confidence: 99%

“…This stabilization problem is challenging due to the severe underactuation of the robot and the highly dynamic nature of the planned trajectories. We implemented a feedback controller based on the method of transverse linearization which has recently emerged as an enabling technology for this type of control problem (Hauser and Chung 1994;Shiriaev et al 2008;Manchester et al 2009;Manchester 2010).…”

Section: The International Journal Of Robotics Research 00(000)mentioning

confidence: 99%

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