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Multibody dynamics
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Applications
• Human and animal motion
• Robotics control
• Hair
• Plants
• Molecular motion
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• Generalized coordinates
• Virtual work and generalized forces
• Lagrangian dynamics for mass points
• Lagrangian dynamics for a rigid body
• Lagrangian dynamics for a multibody system
• Forward and inverse dynamics
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Representations
Maximal coordinates Generalized coordinates
Assuming there are m links and n DOFs in the articulated body, how many constraints do we need to keep links connected correctly in maximal coordinates?
(x0,R0)
(x1,R1)
(x2,R2)
state variables: 18
θ1, φ1
θ2
state variables: 9
x, y, z, θ0,φ0,ψ0
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Maximal coordinates
• Direct extension of well understood rigid body dynamics; easy to understand and implement
• Operate in Cartesian space; hard to
• evaluate joint angles and velocities
• enforce joint limits
• apply internal joint torques
• Inaccuracy in numeric integration can cause body parts to drift apart
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Generalized coordinates
• Joint space is more intuitive when dealing with complex multi-body structures
• Fewer DOFs and fewer constraints
• Hard to derive the equation of motion
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• Generalized coordinates are independent and completely determine the location and orientation of each body
Generalized coordinates
articulated bodies: θ0,φ0,ψ0
θ1, φ1
θ2
x, y, z,
x, y, z, θ,φ,ψone rigid body:
one particle: x, y, z
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Peaucellier mechanism
• The purpose of this mechanism is to generate a straight-line motion
• This mechanism has seven bodies and yet the number of degrees of freedom is one
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• Generalized coordinates
• Virtual work and generalized forces
• Lagrangian dynamics for mass points
• Lagrangian dynamics for a rigid body
• Lagrangian dynamics for a multibody system
• Forward and inverse dynamics
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Virtual work
δri =∂ri
∂q1
δq1 +∂ri
∂q2
δq2 + . . . +∂ri
∂qn
δqn
Fiδri = Fi
!
j
∂ri
∂qj
δqj
ri = ri(q1, q2, . . . , qn)
Represent a point ri on the articulated body system by a set of generalized coordinates:
The virtual displacement of ri can be written in terms of generalized coordinates
The virtual work of force Fi acting on ri is
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Generalized forces
Qj = Fi ·∂ri
∂qj
Define generalized force associated with coordinate qj
virtual work =X
j
Qj�qj
�2
�1
F
M1
M2
Example: l1
l2
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Quiz
✓ ✓ ✓ ✓
Consider a hinge joint theta. Which one has zero generalized force in theta?
(B)(A) (C) (D)
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• Generalized coordinates
• Virtual work and generalized forces
• Lagrangian dynamics for mass points
• Lagrangian dynamics for a rigid body
• Lagrangian dynamics for a multibody system
• Forward and inverse dynamics
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D’Alembert’s principle
• Consider one particle in generalized coordinates under some applied force
!
!
• Applied force and inertia force are balanced along any virtual displacement
firi
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Lagrangian dynamics
• Equations of motion for one mass point in one generalized coordinate
• Ti: Kinetic energy of mass point ri
• Qij: Applied force fi projected in generalized coordinate qj
• For a system with n generalized coordinates, there are n such equations, each of which governs the motion of one generalized coordinate
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Vector form
• We can combine n scalar equations into the vector form
!
!
• Mass matrix:
!
• Coriolis and centrifugal force:
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• Generalized coordinates
• Virtual work and generalized forces
• Lagrangian dynamics for mass points
• Lagrangian dynamics for a rigid body
• Lagrangian dynamics for a multibody system
• Forward and inverse dynamics
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Newton-Euler equations
• There are infinitely many points contained in each rigid body, how do we derive Lagrange’s equations of motion?
• Start out with familiar Newton-Euler equations
!
!
• Newton-Euler describes how linear and angular velocity of a rigid body change over time under applied force and torque
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where
Jacobian matrix
• To express in Lagrangian formulation, we need to convert velocity in Cartesian coordinates to generalized coordinates
• Define linear Jacobian, Jv
!
• Define angular Jacobian, Jω
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Quiz
q1 q2
q3
q4
What is the dimension of the Jacobian?
Which elements in the Jacobian are zero?
q1 q2
q3
q4
x x
(A) (B)
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Lagrangian dynamics
• Substitute Cartesian velocity with generalized velocity in Newton-Euler equations using Jacobian matrices
where,
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Lagrangian dynamics• Projecting into generalized coordinates by multiplying
Jacobian transpose on both sides
!
!
• This equation is exactly the vector form of Lagrange’s equations of motion
where,
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• Generalized coordinates
• Virtual work and generalized forces
• Lagrangian dynamics for mass points
• Lagrangian dynamics for a rigid body
• Lagrangian dynamics for a multibody system
• Forward and inverse dynamics
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Multibody dynamics
• Once Newton-Euler equations are expressed in generalized coordinates, multibody dynamics is a straightforward extension of a single rigid body
!
!
!
!
!
• The only tricky part is to compute Jacobian in a hierarchical multibody system
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Notations
• p(k) returns index of parent link of link k
• n(k) returns number of DOFs in joint that connects link k to parent link p(k)
• Rk is local rotation matrix for link k and depends only on DOFs qk
• R0k is transformation chain from world to local frame of link k
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Jacobian for each link
• Define a Jacobian for each rigid link that relates its Cartesian velocity to generalized velocity of entire system
• Define linear Jacobian for link k
!
• Define angular Jacobian for link k
where
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Example
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• Generalized coordinates
• Virtual work and generalized forces
• Lagrangian dynamics for mass points
• Lagrangian dynamics for a rigid body
• Lagrangian dynamics for a multibody system
• Forward and inverse dynamics
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• Same equations of motion can solve two problems
!
!
• Forward dynamics
• given a set of forces and torques on the joints, calculate the motion
• Inverse dynamics
• given a description of motion, calculate the forces and torques that give rise to it
Forward vs inverse dynamics
M(q)q̈ + C(q, q̇) = Q
q̈ = �M(q)�1(C(q, q̇)�Q)
Q = M(q)q̈ + C(q, q̇)
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Quiz
• Which problem is inverse dynamics?
• Given the current state of a robotic arm, compute its next state under gravity.
• Given desired joint angle trajectories for a robotic arm, compute the joint torques required to achieve the trajectories.
• Given the desired position for a point on a robotic arm, compute the joint angles of the arm to achieve the position.