Advanced Programming for 3D Applications CE Bob Hobbs Staffordshire university Human Motion Lecture 3

2 End Effector 22 a2a2 d 2 =EF-J 2 a 2 x d 2 - Compute instantaneous effect of each joint - Linear approximation to curvilinear motion - Find linear combination to take end effector towards goal position Inverse Jacobian Method

3 Instantaneous linear change in end effector for ith joint = (EF - J i ) x a i

4 Inverse Jacobian Method What is the change in orientation of end effector induced by joint i that has axis of rotation a  i and position J i ? Angular velocity

5 Solution only valid for an instantaneous step Angular affect is really curved, not straight line Once a step is taken, need to recompute solution Inverse Jacobian Method

6 - Mathematics Set up equations y i : state variable x i : system parameter f i : relate system parameters to state variable

7 Inverse Jacobian Method - Mathematics Matrix Form

8 Inverse Jacobian Method - Mathematics Use chain rule to differentiate equations to relate changes in system parameters to changes in state variables

9 Matrix Form Inverse Jacobian Method - Mathematics

10 Inverse Jacobian Method Change in position (and orientation) of end effector Change in joint angles Linear approximation that relates change in joint angle to change in end effector position (and orientation)

11 Inverse Jacobian Method

12

13 = (S - J 1 ) x a 1 =  1

14 The Matrices

15 V – desired linear and angular velocities J – Jacobian Matrix of partials  – change to joint angles (unknowns) 3x1, 6x1 3xN, 6xN N DoFs N x 1

16 Use to bias to desired mid-angle Does not enforce joint angles Does not address “human-like” or “natural” motion Control Term Only kinematic control – no forces involved

17 Jacobian transpose Alternate Jacobian – use goal position HAL – human arm linkage Other ways to numerically IK CCD Damped Least Squares

18 IK w/ constraints Chris Welman, “Inverse Kinematics and Geometric Constraints for Articulated Figure Manipulation,” M.S. Thesis, Simon Fraser University, Basic idea: Constraints are geometric, e.g., point-to-point, point- to-plan, specific orientation, etc. Assume starting out in satisfied configuration Forces are applied to system Detect, and cancel out, force components that would violate constraints.

19 Implementation Handles on skeletons Point handle orientation handle Center-of-mass handle Each handle must know how to compute the Jacobian. Each handle must know how to its value from q

20 Constraints on handles Constraining a point handle to a location Constraining a point handle to a plane Constraining a point handle to a line Constraining an orientation handle to an orientation.

21 Locomotion Tracker has a limited range Tracker has a limited range Must use locomotion metaphor to move greater distances Must use locomotion metaphor to move greater distances Locomotion is on an even plane, virtual terrain may not be even Locomotion is on an even plane, virtual terrain may not be even Collision detection can be employed to raise or lower the participant accordingly Collision detection can be employed to raise or lower the participant accordingly

22 Fly in direction of aim Fly in direction of pointing Fly in direction of gaze Fly in direction of torso Directions of locomotion

 Locomotion Refers to the way a robot moves from place to place. One of the biggest challenges for the robot’s brain.  Effectors and Actuators used for locomotion Legs, wheels, arms, legs, flippers, etc. Legs are more difficult to control than wheels – why? Locomotion

Legged Mobile Robots  Basic advantages of walking single point of contact, amenable to rough terrain, can pick point of contact  Stability Non technical definition: will not tip over - can maintain balance  Static stability motion not required- but requires sufficient number of legs or wheels to accomplish it or – use of brain power to accomplish static stability

Legged Mobile Robots  Statically Stable Walking Being able to walk while remaining stable at all times  Dynamic Stability If a body has dynamic stability, it must move to remain stable. A one-legged hopping robot is dynamically stable, humans are dynamically stable walkers  Center of Gravity The point in or near a body at which the gravitational potential energy of the body is equal to that of a single particle of the same mass located at that point and through which the resultant of the gravitational forces on the component particles of the body acts. Locomotion

Legged Mobile Robots  Gait A sequence of lift and release events for individual legs. It is the particular way a robot (or a legged animal) moves, including the order in which it lifts and lowers its legs and places its feet on the ground.  N = number of distinct event sequences for walking robot with k legs N = (2*k - 1)! k = 2 (humans, etc.): N = (2*2 - 1)! = 3! = 6 Locomotion

Legged Mobile Robots  Gait for 2-legged human (6 event sequences) 1. Both legs down – right down/left up – both legs down 2. Both legs down – right leg up/left leg down – both legs down 3. Both legs down – both legs up – both legs down 4. Right leg down/left leg up – right leg up/left leg down – right leg down/left leg up 5. Right leg down/left leg up – both legs up – right leg down/left leg up 6. Right leg up/left leg down – both legs up – right leg up/left leg down Locomotion

Legged Mobile Robots  Desirable Robot Gaits Have the Following Properties 1. Stability: The robot will not tip over 2. Speed: The robot can move quicikly 3. Energy Efficiency: The robot does not use a lot of energy to move 4. Robustness: The gait can recover from some types of failures 5. Simplicity: The controller for generating the gait is not unwieldy  Some Design Tradeoffs 1. Safety considerations and energy conservation 2. Robustness an simplicity 3. Speed and stability 4. Etc. Locomotion

Legged Mobile Robots  Popularity of 6-Legged Robots  Robust  Allow for multiple gaits Tripod gate (satisfies most of the desirable properties Alternating tripod gate  Robots with more than 6 legs  Ripple gate  Why most robots have 6 legs or wheels  To simplify locomotion Locomotion

Wheeled Robots  Wheels are more efficient than legs – so why not found in nature? Or are they?  Advantages of Wheels (what are diadvantages compared to legs?)  Comparative simplicity of control  Efficiency  Built to be statically stable  Differential Drive Being able to control wheels separately  Differential Steering Ability to steer wheels independently Locomotion