Transcript Dynamics
Dynamics Motion with Regard to Mass Particle Dynamics Mass concentrated in point Newton’s Equation Governs Motion f=Mx Rigid Body Dynamics Two equations govern motion: Newton’s Equation for Translations F= M x Euler’s Equation for Rotational Motion T=Iw+wIw where I is the interial tensor that describes the distribution of mass Dynamics of Links Axis i+1 Newton’s Equation Axis i fi+1 Fi fi+1 Dynamics of Links Axis i+1 Euler’s Equation Axis i ti+1 fi+1 ti fi+1 Ti Bodies in space Conservation of Momentum 0=Mx A body in motion remains in motion Conservation of Angular Momentum 0=Iw The relationship between angular momentum and orientation is tricky Making Them Move In the real world, we do not directly control the kinematic properties of object. We indirectly control position, velocity, and acceleration by exerting forces and torques Current position f Ground Desired position Controllers What force should we apply to move the box to the destination? Current position f Ground Desired position Proportional Control A control law, function, or algorithm for computing forces (or torques). Force is proportional to distance to goal: F = Kp ( xd – x) Workhorse of robotics and animation Problem with Proportional Control Overshoot Goal xd x x time Solution: Damping Proportional Derivative Controller F = Kp ( xd – x) – Kv x virtual friction xd x x time Problem: How Much Damping? Too little damping leads to overshoot x xd time Too much damping leads to sluggishness xd x time Critical Damping Constrain the constants such that: Kv 2 - 4Kp = 0 Just right: No overshoot Fastest possible approach (given gain Kp) There’s always something else What about considering other forces (such as gravity)? desired position F= M x - G G The PD controller will converge to a point where Kp(xd-x) = G current position F? PID Control Proportional Integral Derivative Control F = Kp ( xd – x) – Kv x + Ki ( xd – x) dt