Introduction to Computer Vision and Robotics: Motion Generation Tomas Kulvicius Poramate Manoonpong 1

Motion Control: Trajectory Generation

Different robots –> different motions -> different trajectories 3

How do we generate/plan trajectories? Depends on -what kind of trajectories we need -apllication 4

Movements Movement overview Point-to-Point Periodic Splines DMPs DMPs RNNs NOs GMMs

Movements Overview Point-to-Point Periodic Splines DMPs DMPs RNNs NOs GMMs

Polynomial interpolation Example trajectory sampled by blue points

Polynomial interpolation Sampled trajectory 4th order polynomial Insufficient fit!

Polynomial interpolation Sampled trajectory 6th order polynomial Insufficient fit!

Polynomial interpolation 9th order polynomial Sampled trajectory Runge’s phenomenon

Runge’s phenomenon Runge function 5-th order polyn. 9-th order polyn.

Spline interpolation Idea: many low order polynomials joined together Sampled trajectory Cubic spline No oscillations as compared to polynomial interpolation One can add desired velocity (cubic) or acceleration (5th order) at the end points

Movements Overview Point-to-Point Periodic Splines DMPs DMPs RNNs NOs GMMs

Dynamic Movement Primitives (DMPs)? “DMPs are units of actions that are formalized as stable nonlinear attractor systems” (Ijspeert et al., 2002, Schaal et al., 2003, 2007)

Formalism of discrete DMPs Position change (velocity): Velocity change (acceleration): (v) Exponential decay: g – goal t – temp. scal. Nonlinear function: A set of differential Eqs, which defines a vector field that takes you from any start-point to the goal Kernels: Ijspeert et al., 2002; Schaal et al., 2003, 2007

Formalism of discrete DMPs Position change (velocity): Velocity change (acceleration): (v) Exponential decay: g – goal t – temp. scal. Nonlinear function: Kernels: Ijspeert et al., 2002; Schaal et al., 2003, 2007 Time

DMP properties: 1. Generalization DMPs can be scaled -in time and -space without losing the qualitative trajectory appearance

DMP properties: Position scaling

DMP properties: Generalization DMPs can be scaled in time and space without losing the qualitative trajectory appearance

DMP properties: 2. Robustness to perturbations Real-time trajectory generator – can react to perturbations during movement

DMP properties: 3. Coupling DMPs allow to add coupling terms easily: Temporal coupling Spatial coupling

DMP properties: Temporal coupling Velocity change (acceleration): (v) Exponential decay (phase variable): +Ct Adding additional term Ct allows us to modify the phase of the movement, i.e., stop the movement in case of perturbations.

DMP properties: Phase stopping DMPs are not directly time dependent (phase based) which allows to control phase of the movement (e.g., phase stopping) Without phase stopping With phase stopping

Temporal coupling: Movement stopping

Proactive behavior in humans

What about robots?

DMP properties: Spatial coupling Velocity change (acceleration): +Cs Adding additional term Cs allows us to modify trajectory online by taking sensory information into account, i.e. online obstacle avoidance.

Spatial coupling: Obstacle avoidance

Spatial coupling: Human-Robot interaction

Spatial coupling: Robot-Robot interaction

Comparison of discrete movement generators Method Property Splines DMPs GMMs Time dependence Direct Indirect Inde- pendent Robustness to perturbations No Yes Generalization Set of trajectories

Movements Overview Point-to-Point Periodic Splines DMPs DMPs RNNs NOs GMMs

Formalism of discrete DMPs: Reminder Position change (velocity): Velocity change (acceleration): (v) Exponential decay: g – goal t – temp. scal. Nonlinear function: Kernels: Ijspeert et al., 2002; Schaal et al., 2003, 2007 Time

Formalism of rhythmic DMPs Position change (velocity): Velocity change (acceleration): (f,A) Limit cycle oscillator with constant phase speed: g – baseline A – amplitude t – frequency Nonlinear function: Kernels: Ijspeert et al., 2002; Schaal et al., 2003, 2007 Time

Movements Overview Point-to-Point Periodic Splines DMPs DMPs RNNs NOs GMMs

Neural oscillators Central Pattern Generator (CPG) Pattern generation without sensory feedback (Open-loop system) 36

Dynamical system approach: Van der Pol Oscillator CPG methods Dynamical system approach: Van der Pol Oscillator Dynamic Movement Primitives Neural control approach: Matsuoka Oscillator 2-neuron Oscillator 37

2-neuron oscillator Neural structure: 2-neuron network [Pasemann et al., 2003] Central pattern generator (CPG): Self excitatory + excitatory & inhibitory synapses The activation function The transfer function

2-neuron oscillator W12 = - W21 W11, W22 Pasemann, F., Hild, M., Zahedi, K. SO(2)-Networks as Neural Oscillators, Mira, J., and Alvarez, J. R., (Eds.), Computational Methods in Neural Modeling, Proceedings IWANN 2003, LNCS 2686, Springer, Berlin, pp. 144-151, 2003.

CPG with modulatory input

Different walking gates (AMOS II)

Movements Overview Point-to-Point Periodic Splines DMPs DMPs RNNs NOs GMMs

Reflexive neural networks Reflexes - local motor response to a local sensation Locomotion as a chain of reflexes: purely sensory-driven system (Closed-loop system).

Reflexive neural network: application to bipedal robot RunBot each hip or knee joint of the robot has two states of movement: extension and flexion 44

Sensor-triggered generation of movement 1) Left leg touches the ground: GL = active  Left hip flexes (backward) & Left knee extends (straight) = STANCE  Right hip extends (forward) & Right knee flexes (bend) = SWING 2,3) Right Hip angle reaches AEA (Anterior Extreme Angle) AEA = active  Right knee extends (straight) Right leg still in swing, Left leg still in stance 4,5) Right leg touches the ground: GR = active  Right hip flexes (backward) & Right knee extends (straight) = STANCE  Left hip extends (forward) & Left knee flexes (bend) = SWING each hip or knee joint of the robot has two states of movement: extension and flexion GL 45

Reflexive neural network of RunBot each hip or knee joint of the robot has two states of movement: extension and flexion 46

Passive dynamic walking 47

Learning to walk up a ramp Neural learning 48

RunBot learning to walk up a ramp 49

Reflex based methods Pros: • Very close link between the controller and what the robot actual does Cons: • because of the lack of a centrally generated rhythm, locomotion might be completely stopped because of damage in the sensors and/or external constraints that force the robot in a particular posture.

Comparison of periodic movement generators Method Property DMPs Neural Oscillators RNNs Time dependence Indirect Direct Robustness to perturbations Yes Generalization Yes&No N/A Arbitrary trajectory

Joining movement sequences: human vs. robot We want to achieve human like motions – smooth transitions between consequent movements.

Formalism of original DMPs: Reminder Position change (velocity): Velocity change (acceleration): ( ) Delayed goal: Exponential decay: Nonlinear function: Time Kernels: Ijspeert et al., 2002; Schaal et al., 2003, 2007

Modification of original DMP’s Goal function: Sigmoidal decay: Nonlinear function:

Joining DMPs by using overlapping kernels Goal: to join accurately in position and velocity space at the joining point at the specific time T (provided by human example)

Comparison: orig. DMPs vs. novel approach Joining letters “a” and “b” Sequential joining Dt Sequential joining Novel approach

Joining demo: Handwriting

Joining demo: joining of discrete and repetitive movements

Summary Types of motions: - discrete (poin-to-point); - oscillatory (repetitive). Movement generation frameworks: - splines; - dynamic movement primitives (DMPs); neural oscillators (NOs); reflexive neural networks (RNNs). There is no best trajectory generator – it much depends on the application!