- The document summarizes the gait pattern modulator and trajectory generator for the Cheetah 1 robot. - It describes how phase signals are used to impose gait patterns and synchronize legs based on touch down events. The phase signals change to enable transitions between trots and gallops. - The trajectory generator designs swing phase trajectories using Bezier curves defined by control points. Stance phase trajectories were originally sinusoidal waves but are now controlled by force in Cheetah 3.

1. Review
MIT Cheetah 1
: Gait-Pattern Modulator, Trajectory Generator[1]
ModuLabs
강남Dynamics Lab
Hancheol Choi
(babchol@gmail.com)

2. Contents
Focusing only gait pattern and trajectory generator in Cheetah 3
- Gait pattern modulator
- Phase signals
- Synchronization by proprioceptive sensory feedback ( Contact Model Fusion)
- Imposing gait-patterns
- Phase signal generation according to TD elapsed time and phase differences
- Change of gait-pattern for trot-to-gallop gait transition
- Leg trajectory generator
- Design of the trajectory in the swing phase
- Design of the trajectory in the stance phase ( force control using QP or MPC in Cheetah 3)

3. Gait pattern modulator
Phase signal, ~ [0, 1] : normalized timed location in the desired stride period
Stride time period:
- Swing period is relatively constant over a wide range of speeds [Maes et al., 2008]
 try to minimize the swing phase and maximize the ground phase
 improve stability and minimize peak torque of the motor
 (0.25 s)
- Stride frequency and contact time ↓ as speed ↑ [Vilensky et al.,1991, Wickler et al.,2003]

Gait pattern
- Phase signal, on reference leg (ex. Front Left)
- Phase lag, on the other legs
: half of the stroke length
: desired speed

4. Synchronization by proprioceptive sensory feedback
Touch down(TD) event, Lift off(LO) event is important in locomotion.
TD Event triggers the start of the gait modulator and synchronize it with the environment.
Force sensor at the foot can find TD Event.
But, torque-controlled leg can find TD Event with the proprioceptive sensory feedback.
In Cheetah 1
- Abrupt change in Radial force of leg  TD or LO event
- Block multiple mis-detections by disabling full stance phase and 90 % of swing phase after
detecting TD once
In Cheetah 3
- Developed into contact model fusion using discrete momentum observer and kalman filter
[Geraldo Bledt et all., 2018]
TD Event based “stride-to-stride” pattern modulation
- TD triggers reference leg’s phase signal,
- The other legs are following with phase lag,
- And wait until another TD event detected  undesired stationary period, but this is minimal
adaptation to the environment

5. Phase signal generation(1)
Elapsed time after TD event
Phase signal of stance phase and swing phase
Phase lag is determined by a specific gait-pattern.
After 90% swing phase, and TD event occured

6. Phase signal generation(2)
Gait transition from trot to gallop: linear transition
SAT is saturation function

7. Leg trajectory generator
Swing: defined from Bezier curve defined by a set of 12 control points
Stance: sinusoidal waves  substituted by GRF control

8. Design of the Trajectory in Swing-Phase
Parameterized by a corresponding swing-phase signal,
Pri
 control velocity by distance between control points intuitively
Principals of y-direction design Principals of x-direction design
Principals:
- double overlapped control points make velocity 0
- tripple overlapped control points make acceleration 0

9. Control Points Table in Swing-Phase
Ground clearance = 150mm/s
Swing-leg retraction speed= 3.86m/s, @
Angle of attack = 68 deg
Can be scaled by

10. Appendix. Bezier curves (1)
Control points P0, … Pn, where n is called its order (n=1, lienar, n=2, quadratic, …)
 represent curves in graphics
Idea
1. Linear Bezier curves
2. Quadratic Bezier curves
Derivation, interpreted as the linear interpolant of n-1 order(linear) bezier curves
Re-arrange for the control points
Re-arrange highlighting symmetry with respect to P1
First derivatives
Second derivatives
P0
P1 P2

11. Appendix. Bezier curves (2)
Generalization of Bezier curves
Explicit definition

12. Reference
[1] Dong Jin Hyun, Sangok Seok, Jongwoo Lee and Sangbae Kim, High speed trot-
running: Implementation of a hierarchical controller using proprioceptive
impedance, IJRR, 2014.
[2] Vilensky JA, Libii JN and Moore AM (1991) Trot–gallop gait transitions in
quadrupeds. Physiology and Behavior 50: 835–842.
[3] Maes LD, Herbin M, Hacker R, Bels VL and Abourachid A (2008) Steady
locomotion in dogs: Temporal and associated spatial coordination patterns and
the effect of speed. The Journal of Experimental Biology 211: 138–149.
[4] Bezier curve wikipedia (https://en.wikipedia.org/wiki/B%C3%A9zier_curve)