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R.Parent, CSE788 OSU Constrained Body Dynamics Chapter 4 in: Mirtich Impulse-based Dynamic Simulation of Rigid Body Systems Ph.D. dissertation, Berkeley, 1996

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R.Parent, CSE788 OSU Preliminaries Links numbered 0 to n Fixed base: link 0; Outermost like: link n Joints numbered 1 to n Link i has inboard joint, Joint i Each joint has 1 DoF Vector of joint positions: q=(q 1,q 2,…q n ) T 1 2n …

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R.Parent, CSE788 OSU The Problem Given: –the positions q and velocities of the n joints of a serial linkage, –the external forces acting on the linkage, –and the forces and torques being applied by the joint actuators Find: The resulting accelerations of the joints:

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R.Parent, CSE788 OSU First Determine equations that give absolute motion of all links Given: the joint positions q, velocities and accelerations Compute: for each link the linear and angular velocity and acceleration relative to an inertial frame

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R.Parent, CSE788 OSU Notation Linear velocity of link i Linear acceleration of link i Angular velocity of link i Angular acceleration of link i

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R.Parent, CSE788 OSU Joint variables Unit vector in direction of the axis of joint i vector from origin of F i-1 to origin of F i vector from axis of joint i to origin of F i joint position joint velocity

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R.Parent, CSE788 OSU Basic terms uiui riri didi FiFi F i-1 State vector Axis of articulation Frames at center of mass F i – body frame of link i Origin at center of mass Axes aligned with principle axes of inertia

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R.Parent, CSE788 OSU From base outward Velocities and accelerations of link i are completely determined by: 1. the velocities and accelerations of link i-1 2. and the motion of joint i

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R.Parent, CSE788 OSU First – determine velocities and accelerations From velocity and acceleration of previous link, determine total velocity and acceleration of current link From local joint Computed from base outward To be computed

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R.Parent, CSE788 OSU Compute outward Angular velocity of link i = angular velocity of link i-1 plus angular velocity induced by rotation at joint i Linear velocity:

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R.Parent, CSE788 OSU Compute outward Rewritten, using Angular acceleration propagation Linear acceleration propagation and (from previous slide) (relative velocity)

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R.Parent, CSE788 OSU Define w rel and v rel and their time derivatives prismatic revolute Axis times parametric velocityAxis times parametric acceleration Joint velocity vector Joint acceleration vector

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R.Parent, CSE788 OSU Velocity propagation formulae (revolute) linear angular

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R.Parent, CSE788 OSU Time derivatives of v rel and w rel (revolute) Joint acceleration vector Change in joint velocity vector From change in joint velocity vector From joint acceleration vector From change in change in vector from joint to CoM

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R.Parent, CSE788 OSU Derivation of (revolute)

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R.Parent, CSE788 OSU Propagation formulae (revolute) linear angular

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R.Parent, CSE788 OSU First step in forward dynamics Use known dynamic state (q, q-dot) Compute absolute linear and angular velocities Acceleration propagation equations involve unknown accelerations But first – need to introduce notation to facilitate equation writing Spatial Algebra

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R.Parent, CSE788 OSU Spatial Algebra Spatial velocity Spatial acceleration

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R.Parent, CSE788 OSU Spatial Transform Matrix r – offset vectorR– rotation (cross product operator)

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R.Parent, CSE788 OSU Spatial Algebra Spatial force Spatial transpose Spatial inner product Spatial joint axis (used in later)

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R.Parent, CSE788 OSU ComputeSerialLinkVelocities For i = 1 to N do end R rotation matrix from frame i-1 to i r radius vector from frame i-1 to frame i (in frame i coordinates) (revolute) Specific to revolute joints

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R.Parent, CSE788 OSU Spatial formulation of acceleration propagation (revolute) Want to put in form: Previously:

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R.Parent, CSE788 OSU Spatial Coriolis force (revolute) These are the terms involving

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R.Parent, CSE788 OSU Featherstone algorithm Spatial acceleration of link i Spatial force exerted on link i through its inboard joint Spatial force exerted on link i through its outboard joint All expressed in frame i Forces expressed as acting on center of mass of link i

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R.Parent, CSE788 OSU Serial linkage articulated motion Spatial articulated inertia of link i Spatial articulated zero acceleration force of link I (independent of joint accelerations) Develop equations by induction

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R.Parent, CSE788 OSU Base Case Force/torque applied by inboard joint + gravity = inertia*accelerations of link Consider last link of linkage (link n) Newton-Euler equations of motion

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R.Parent, CSE788 OSU Using spatial notation Inboard joint Link n

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R.Parent, CSE788 OSU Inductive case Inboard joint outboard joint Link i-1

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R.Parent, CSE788 OSU Inductive case The effect of joint I on link i-1 is equal and opposite to its effect on link i Substituting…

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R.Parent, CSE788 OSU Inductive case Invoking induction on the definition of

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R.Parent, CSE788 OSU Inductive case Express a i in terms of a i-1 and rearrange Need to eliminate from the right side of the equation

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R.Parent, CSE788 OSU Inductive case Magnitude of torque exerted by revolute joint actuator is Q i Moment of force d f u A force f and a torque applied to link i at the inboard joint give rise to a spatial inboard force (resolved in the body frame) of

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R.Parent, CSE788 OSU Inductive case Premultiply both sides bysubstitute for Q i, and solve previously and

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R.Parent, CSE788 OSU And substitute

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R.Parent, CSE788 OSU And form I & Z terms To get into form:

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R.Parent, CSE788 OSU Ready to put into code Using Loop from inside out to compute velocities previously developed (repeated on next slide) Loop from inside out to initialization of I, Z, and c variables Loop from outside in to propagate I, Z and c updates Loop from inside out to compute q-dot-dot using I, Z, c

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R.Parent, CSE788 OSU ComputeSerialLinkVelocities For i = 1 to N do end R rotation matrix from frame i-1 to i r radius vector from frame i-1 to frame i (in frame i coordinates) (revolute) Specific to revolute joints

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R.Parent, CSE788 OSU InitSerialLinks For i = 1 to N do end (revolute)

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R.Parent, CSE788 OSU SerialForwardDynamics For i = n to 2 do Call compSerialLinkVelocities Call initSerialLinks For i = 1 to n do

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