ROBOT DYNAMICS(Differential Motion Of Frame, Jacobian,

Static Force Analysis Of Robot)By

R.C.Sainiwww.rcsaini.blogspot.com

Differential Relationship• Consider simple mechanism with

2-degrees of freedom

• Each link can rotate independently

• Rotation of link 1 (θ1), relative to reference frame

• 2nd link rotation (θ2), relative to link 1

• Velocity of point B :

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• Writing the velocity equation in matrix form :

• Velocity relationship by differentiating the equation that describe the position of point B :

• Differentiating with respect to θ1, θ2

• In matrix form

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JACOBIAN

• Representation of geometry of the element of the mechanism in time

• Its allow conversion of differential motion of individual joints to differential motion of point of interest

• Also relate individual joint motion to overall mechanism

• It is time related• θ1, θ2 vary in time• As per earlier slide – Jacobian was formed by

differentiated position equations with respect to θ1, θ2

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• Suppose that we have a set of equation Yi in terms of a set of variables Xj

• Differentiating

• In matrix

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JACOBIAN Cont.• Or

• dx, dy and dz – differential motion of the hand along x, y and z axis• δx, δy and δz – differential rotation of the hand around x, y and z axis• Dθ – differential motion of the joint

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Differential Motion Of Frame

• Dived into 3 types :-– Differential Translation– Differential Rotation– Differential Transformation (translation + rotation)

• Differential Translation dx, dy and dz with respect to x, y and z axis

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Differential Motion Of Frame• Differential Rotation– δx , δy and δz are differential rotation about x, y and

z axis respectively– Use following approximation :

– So ,

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Differential Motion Of Frame• Differential Rotation– Rotation about a general axis “k”Rot(k, dθ)=Rot(x, dθ) X Rot(y, dθ) X Rot(z, dθ)

- Neglecting higher order differential

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Differential Motion Of Frame• Differential Transformation (translation + rotation)

• Original frame = T• Differential transformation = dT[T+dT] = [Trans(dx,dy,dz) Rot(k, dθ)] [T][dT] = [Trans(dx,dy,dz) Rot(k, dθ) – I][T] , {I = Unit Matrix}Δ = Trans(dx,dy,dz) Rot(k, dθ) – IΔ = Differential Operator

Δ =

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Differential Changes Between Frames

• Δ represents a differential operator relative to fixed reference frame and it technically u Δ

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Static Force Analysis Of Robot

• Robot may be under either position control or force control

• E.g. – Robot follow prescribes path to cut groove in flat surface, it is under position control. However, if the surface has a slight unknown curvature in it, but robot following a given path, it will groove more deeper or less deeper on surface. Alternatively, suppose that the robot were to measure the force of exerting on the surface while cutting the groove. Now robot adjust the depth by joint+link movement for uniformly cut, now it is under force control.

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Static Force Analysis Of Robot• To relate the joint forces and torques to forces and moments generated at

the hand frame of the robot, we define

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Static Force Analysis Of Robot

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THANK YOU

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