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

2. Motion Control: Trajectory Generation

3. Different robots –> different motions -> different trajectories3

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

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

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

7. Polynomial interpolation
Example trajectory sampled by blue points

8. Polynomial interpolation
Sampled trajectory
4th order polynomial
Insufficient fit!

9. Polynomial interpolation
Sampled trajectory
6th order polynomial
Insufficient fit!

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

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

12. 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

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

14. 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)

15. 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

16. 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

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

18. DMP properties: Position scaling

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

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

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

22. 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.

23. 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

24. Temporal coupling: Movement stopping

25. Proactive behavior in humans

26. What about robots?

27. 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.

28. Spatial coupling: Obstacle avoidance

29. Spatial coupling: Human-Robot interaction

30. Spatial coupling: Robot-Robot interaction

31. 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

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

33. 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

34. 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

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

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

37. 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

38. 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

39. 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 , 2003.

40. CPG with modulatory input

41. Different walking gates (AMOS II)

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

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

44. 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

45. 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

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

47. Passive dynamic walking47

48. Learning to walk up a ramp
Neural learning 48

49. RunBot learning to walk up a ramp49

50. 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.

51. 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

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

53. 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

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

55. 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)

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

57. Joining demo: Handwriting

58. Joining demo: joining of discrete and repetitive movements

59. 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!