Dynamic model of robot manipulators
Claudio Melchiorri
Dipartimento di Elettronica, Informatica e Sistemistica (DEIS)
Universit`a di Bologna
email: [email protected]
C. Melchiorri (DEIS) Dynamic Model 1 / 65
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Dynamic model of robot manipulators, Lecture notes of Introduction to Robotics
Dynamic model of robot manipulators Claudio Melchiorri, Dipartimento di Elettronica, Informatica e Sistemistica (DEIS), Universit`a di Bologna
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2016/2017
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Dynamic model of robot manipulators
Claudio Melchiorri
Dipartimento di Elettronica, Informatica e Sistemistica (DEIS) Universit`a di Bologna email: [email protected]
Summary
(^1) Dynamic models Introduction Euler-Lagrange model
Dynamic models Introduction
Dynamic model of manipulators
Normally, a manipulator is composed by an open kinematic chain, and its dynamic model is affected by several “drawbacks”: low rigidity (elasticity in the structure and in the joints) potentially unknown parameters (dimensions, inertia, mass,... ) dynamic coupling among links.
Other non linear effects are usually introduced by the actuation system: friction dead zones
...
In any case, in the derivation of the dynamic model, an ideal case of a series of connected rigid bodies is made.
Dynamic models Introduction
Dynamic model of manipulators
Two problems may be defined for the study of the dynamic model:
Direct dynamic model: computation of the time evolution of ¨q(t) (and then of q(t), q˙(t)), given the vector of generalized forces (torques and/or forces) bτ (t) applied to the joints and, in case, the external forces applied to the end-effector, and the initial conditions q(t = t 0 ), q˙(t = t 0 ).
τ (t) =⇒ ¨q(t) ( ˙q(t), q(t) )
Inverse dynamic problem: computation of the vector τ (t) necessary to obtain a desired trajectory ¨q(t), q˙(t), q(t), once the forces applied of the end-effector are known.
¨q(t), q˙(t), q(t) =⇒ τ (t)
Euler-Lagrange model
Generalized variables, or Lagrange coordinates: Independent variables used to describe the position of rigid bodies in the space.
For the same physical system, more choices for the Lagrangian coordinates are usually possible.
In robotics =⇒ joint variables q 1 , q 2 ,... qn.
Euler-Lagrange model
From physics, we know that it is possible to define: (^1) The Kinetic Energy function K (q, q˙) (^2) The Potential Energy function P(q) and therefore the Lagrangian function L(q, q˙) = K (q, q˙) − P(q)
The Euler-Lagrange equations are
ψi =
d dt
∂L
∂ q˙i
∂L
∂qi i = 1,... , n
being ψi the non-conservative (external or dissipative) generalized forces performing work on qi. In robotics, we consider:
τi joint actuator torque [ JT^ Fc
] i term due to external (contact) forces dii q˙i joint friction torque
Therefore: ψi = τi +
[
JT^ Fc
]
i −^ dii^ q˙i^.
Euler-Lagrange model
Computation of the Kinetic Energy. For a rigid body:
The mass can be computed by
m =
B
ρ(x, y , z) dx dy dz
where ρ(x, y , z) is the mass density (assumed constant): ρ(x, y , z) = ρ. The center of mass (CoM) is
pC =
m
B
p(x, y , z)ρ dx dy dz =
m
B
p(x, y , z) dm
The kinetic energy results
K =
B
vT^ (x, y , z)v(x, y , z)ρ dx dy dz
B
vT^ (x, y , z)v(x, y , z) dm
Euler-Lagrange model
Let assume that the translational vC and rotational ω velocities of the center of mass are known with respect to an inertial frame F 0.
The velocity of a generic point p′^ of the body is
v = vC + ω × (p′^ − pC ) = vC + ω × r (2)
The velocity expressed in a frame F’ fixed to the rigid body may be computed by introducing the rotation matrix R betweeen F’ and F 0
RT^ v = RT^ (vC + ω × r) = RT^ vC + (RT^ ω) × (RT^ r)
Therefore v′^ = v′ C + ω′^ × r′
Euler-Lagrange model
In conclusion K =
B
vTC vC dm +
B
rT^ ST^ Sr dm
The first term depends on the linear velocity vC of the center of mass
1 2
B
vTC vC dm =
m vTC vC
For the second term, considering that aT^ b = Tr (a bT^ ) and the particular structure of matrix S, we have
1 2
B
rT^ ST^ Sr dm =
B
Tr (Sr rT^ ST^ ) dm =
Tr (S
B
rrT^ dm ST^ )
Tr (S J ST^ ) =
ωT^ I ω I : body inertia matrix
Also this term depends on the velocity of the center of mass (in this case ω).
Euler-Lagrange model
Matrix J, Euler matrix, and matrix I, body inertia matrix, are symmetric, and have the following general expressions:
J =
r (^) x^2 dm
rx ry dm
∫ rx^ rz^ dm rx ry dm
r (^) y^2 dm
∫ ry^ rz^ dm rx rz dm
ry rz dm
r (^) z^2 dm
I =
(r (^) y^2 + r (^) z^2 ) dm −
rx ry dm −
rx rz dm −
rx ry dm
(r (^) x^2 + r (^) z^2 ) dm −
ry rz dm −
rx rz dm −
ry rz dm
(r (^) x^2 + r (^) y^2 ) dm
The elements of both matrices J ed I depend on vector r, i.e. on the position of the generic point of the i-th link with respect to its center of mass, defined in the base frame.
Since the position of the i-th link depends on the configuration of the manipulator, matrices J and I are in general functions of the joint variables q!
Euler-Lagrange model
In conclusion, the kinetic energy of a rigid body is defined as (Konig Theorem)
K =
m vTC vC +
ωT^ Iω (3)
Both terms depend only on the velocity of the rigid body.
The first term, being related to the magnitude of a vector (‖vC ‖ = vCT vC ), is invariant with respect to the reference frame used to express the velocity: vTC vC = (RvC )T^ (RvC ) = vTC (RT^ R)vC , ∀ R This property holds also for the second term: the product ωT^ Iω is invariant with respect to the reference frame. As a matter of fact, the body inertia matrix is transformed according to the following relation: I′^ = R I RT Then: ωT^ Iω = ω′T^ I′ω′^ = (R ω)T^ (RIRT^ )(Rω) = ωT^ (RT^ R)I(RT^ R)ω.
Therefore, being (3) invariant with respect to the reference frame, F can be chosen in order to simplify the computations required for the evaluation of K.
Euler-Lagrange model
Since the kinetic energy Ki of the generic i-th link is invariant with respect to the reference frame (used to express vC , ω, I), it is convenient to chose a frame Fi fixed to the link itself, with origin in the center of mass. In this manner matrix I is constant and simple to be computed on the basis of the geometric properties of the link. On the other hand, normally the rotational velocity ω is defined in the base frame F 0 , and therefore it is needed to transform it according to RT^ ω, being R the rotation matrix between Fi and F 0 (known from the kinematic model of the manipulator).
Euler-Lagrange model
The end-effector velocity may be computed as a function of the joint velocities ˙q 1 ,... , q˙n through the Jacobian matrix J. The same methodology can be used to compute the velocity of a generic point of the manipulator, and in particular the velocity vCi = ˙pCi of the center of mass pCi , that results function of the joint velocities ˙q 1 ,... , q˙i only: p˙Ci = Jiv 1 q˙ 1 + Jiv 2 q˙ 2 +... + Jivi q˙i = Jiv q˙ ωi = Ji ω 1 q˙ 1 + Ji ω 2 q˙ 2 +... + Ji ωi q˙i = Ji ω q˙
where
Jiv =
[ Jiv 1... Jivi 0... 0
]
Ji ω =
[ Ji ω 1... Ji ωi 0... 0
]
with [ Jivj Ji ωj
]
[ zj− 1 × (pCi − pj− 1 ) zj− 1
] rotational joint [ Jivj Ji ωj
]
[ zj− 1 0
] linear joint
being pj− 1 the position of the origin of the frame associated to the j-th link.
Euler-Lagrange model
In conclusion, for a n dof manipulator we have:
K =
∑^ n
i =
mi vCiT vCi +
∑^ n
i =
ωTi Ri˜Ii RTi ω
q˙T
∑n
i =
[
mi Ji Tv (q)Jiv (q) + Ji T ω (q)Ri˜Ii RTi Ji ω (q)
]
q^ ˙
q˙T^ M(q) ˙q
∑^ n
i =
∑^ n
j=
Mij (q) ˙qi q˙j
where M(q) is a n × n, symmetric and positive definite matrix, function of the manipulator configuration q. Matrix M(q) is called the Inertia Matrix^ of the manipulator.