16-811: Math Fundamentals for Robotics, Fall 2014 Finding minimum energy trajectories of a two linked pendulum

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1 16-811: Math Fundamentals for Robotics, Fall 014 Finding minimum energy trajectories of a two linked pendulum Lerrel Pinto < lerrelp > December 1th 014 Problem Statement: Find a zero energy end effector trajectory for a compound double pendulum. Input: For a preliminary study, let us consider the following End effector initial position (x i,y i ) End effector final position (x f,y f ) Time for execution of trajectory (T) Output: Joint trajectory (Θ(t), Θ(t)) that requires 0 torque to execute i.e simply getting the joints to (Θ(0), Θ(0)) would be sufficient to reach (Θ(T), Θ(T)). 1 Introduction & Motivation Energy expenditure considerations have been an important part of legged robotics community. A breakthrough in this was pioneered by Dr. Tad McGeer in his work on passive dynamic walking[1]. His work sparked interest among several researchers, most notably Andy Ruina s group in Cornell University. An interesting application of this concept is the Cornell Ranger[] that has walked 65Km on one charge of battery. The underlying concept of passive dynamics is tricky to understand. Intuitively, let us ask a question, Why does a ball roll down a slope? ; The answer is gravity which dictates the dynamics of the ball. Similarly, by asking questions like Can we exploit the natural dynamics of complex mechanisms?, Could we make these complex mechanisms move from point A to point B without expending energy whatsoever, one would invariably have to work with passive dynamics. 1

2 The concepts of passive dynamics, being independent of the study of legged robots, can be applied to robot arms[3] and possibly to other mechanical systems. A possible avenue of work could be on the manufacturing floor, where repetitive actions usually performed in a completely active closed loop control modes ignoring passivity. Via the medium of this project and course, I would like to find these minimum energy trajectories in a very non-complete way; By trying to exploit the chaotic behavior of some dynamical systems, I try to simulate the passivity of a system and create a database of passive trajectories. Then given a query input, I search in this database for the most similar previously seen trajectory. In this work, a double pendulum is considered since it is equivalent to a revolute joint robotic arm under the effect of gravity. Figure 1: Compound double pendulum. Courtesy: Andy Ruina Method The method which was briefly explained in the Introduction can be seen in the following two stage pipeline. Stage 1: Trajectory Generator The first step in generating the trajectories is by using the lagrange equations.

3 Figure : Method Pipeline Dynamics: Given the double pendulum in Figure 3. Figure 3: Compound Pendulum Derivation. Courtesy: R. Nielsen L = 0.5I 1 θ I,cm θ + 0.5m h 1 θ m d θ + m h 1 d θ 1 θ cos(θ 1 θ ) + m gd 1 cos(θ 1 ) + m gh 1 cos(θ 1 ) + m gd cos(θ 1 ) (1) 3

4 Now the equations of motion can be derived from this and get the following equations. τ 1 = I 1 θ1 + m h θ m h 1 d θ cos(θ 1 θ ) m h 1 d θ ( θ 1 θ )sin(θ 1 θ ) + m h 1 d θ 1 θ sin(θ 1 θ ) + m 1 gd 1 sin(θ 1 ) + m gh 1 sin(θ 1 ) () τ = I,cm θ + m d θ + m h 1 d θ1 cos(θ 1 θ ) m h 1 d θ 1 ( θ 1 θ )sin(θ 1 θ ) m h 1 d θ 1 θ sin(θ 1 θ ) + m gd sin(θ ) (3) Now since we know τ 1 = τ = 0, We can decouple the two equations by solving in θ 1 and θ. Differential equation solving After decoupling the dynamic equations, the equations are written in state space derivative format, introducing variables θ 1 and θ along with the existing required variables θ 1 and θ. We solve these four simulataneous ODE equations using the ode113 solver on Matlab. The ode113 solver is an Adam Bashforth based multistep solver which works much faster than the ode45 Runge Kutta based solver. Seed generation: Initial states To get a wide variety of trajectories, it is important to seed the trajectory generator with variety of potential energies. This is important as it specifies an upper limit on the kinetic energy of the system at any point of the trajectory. This limit could be determined by the physical constraints on the pendulum (usually the joint motor maximum velocity). How does chaos help? Given an initial state (thus fixing the total energy of the trajectory), simulating the system for a sufficiently long time would give nonrecurring trajectory which generally causes the end effector to span the end effector position space. An illustration of this can be seen in Figure 3. Database of trajectory In the current implementation, no special structures are used to store the trajectory data. This seemed sufficient for a DOF system. However for extensions to multiple DOF systems, a well structured database would be required. Stage : Greedy Search Given an query input, the database is searched for a best match. The distance metric used in evaluating a match is the euclidean distance between the query initial and final end effector points and the database matched initial and final points. 4

5 Figure 4: Chaotic end effector movement 3 Results The results seem quite accurate through all the randomly queried inputs. 3 of these randomly queried inputs along with 3 different values of T are shown in Figure 5. 4 Conclusions Through this class project, Ive developed a novel way of finding minimum cost trajectories and successfully implemented on a double compound pendulum. Future work could involve extensions to higher DOF systems. However before this, a well structured database should be implemented. It would be interesting to implement obstacle avoidance into the matching process and make it possible to specify explicit trajectory constraints like length minimisation. References [1] Tad McGeer. Passive dynamic walking. the international journal of robotics research, 9():6 8, [] JCPA Bhounsule, Jason Cortell, and Andy Ruina. Design and control of ranger: an energy-efficient, dynamic walking robot. In Proc. CLAWAR, pages , 01. [3] Matthew M Williamson. Robot arm control exploiting natural dynamics. PhD thesis, Massachusetts Institute of Technology,

6 Figure 5: Green star signifies the start point. Red star indicates the goal point. The Time for execution specified is 0.5s, 1s and 3s respectively over the columns. 6