ECE5463: Introduction to Robotics Lecture Note 10...

ECE5463: Introduction to Robotics

Lecture Note 10: Generalized Force and Staticsof Open Chains

Prof. Wei Zhang

Department of Electrical and Computer EngineeringOhio State UniversityColumbus, Ohio, USA

Spring 2018

Lecture 10 (ECE5463 Sp18) Wei Zhang(OSU) 1 / 12

Outline

• Wrench

• Statics of Open Chains

Outline Lecture 10 (ECE5463 Sp18) Wei Zhang(OSU) 2 / 12

Wrench

• Consider a rigid body with body frame and consider a force f acting on apoint r on the rigid body

• Define an arbitrary stationary frame {a} and let ra and fa be the {a}-framerepresentations of r and f vectors. This force create a torque or momentma ∈ R3 in frame {a}

ma = ra × fa

• Similar to twist, we can merge the moment and force into a single 6D vector.This vector is called the spatial force or wrench.

Fa =

[ma

fa

]

Wrench Lecture 10 (ECE5463 Sp18) Wei Zhang(OSU) 3 / 12

Wrench-Twist Pair and Power

• Recall that for a point mass with linear velocity v and linear force f . Then weknow that the power (instantaneous work done by f) is given by f · v = fT v

• This relation can be generalized to spatial force (i.e. wrench) and spatialvelocity (i.e. twist)

• Suppose a rigid body has a twist Va = (ωa, va) expressed in {a}, and a forcef is applied at a point r on the rigid body with wrench Fa. Then the power issimply

Va · Fa = VTa Fa = ωTama + vTa fa


Rotational Power

• Consider a point mass with a pure rotational velocity ωa = θωa, and amoment ma, relative to frame {a}

• Our previous discussion indicates that its power is

ωTama = θ · (ωTama) , θ · τ

• τ = ωTama = mTa ωa is the projection of the moment onto the rotation axis,

i.e. the effective part of the moment.

• Often times, τ is also referred to as ”torque” with the understanding that itis a scalar quantifying the effectiveness of a moment (i.e. vector torque)relative to some rotation axis.


Wrench Representations in Different Frames

• The wrench Fa can be expressed in another frame {c}, provided Tac is known

• This is not simply rewriting the coordinates of the vectors m and f in {c}.

• We have to change the vector representation of the point r from ra (vectorfrom the origin of {a} to r, expressed in {a}) to rc (vector from the origin of{c} to r, expressed in {c})

Chapter 3. Rigid-Body Motions 109

{a}

{c}

r

f

ra

rc

Figure 3.21: Relation between wrench representations Fa and Fb.

Just as with twists, we can merge the moment and force into a single six-dimensional spatial force, or wrench, expressed in the {a} frame, Fa:

Fa =

[ma

fa

]∈ R6. (3.93)

If more than one wrench acts on a rigid body, the total wrench on the body issimply the vector sum of the individual wrenches, provided that the wrenchesare expressed in the same frame. A wrench with a zero linear component iscalled a pure moment.

A wrench in the {a} frame can be represented in another frame {b} (Fig-ure 3.21) if Tba is known. One way to derive the relationship between Fa and Fbis to derive the appropriate transformations between the individual force andmoment vectors on the basis of techniques we have already used.

A simpler and more insightful way to derive the relationship between Faand Fb, however, is to (1) use the results we have already derived relatingrepresentations Va and Vb of the same twist, and (2) use the fact that the powergenerated (or dissipated) by an (F ,V) pair must be the same regardless of theframe in which it is represented. (Imagine if we could create power simplyby changing our choice of reference frame!) Recall that the dot product of aforce and a velocity is a power, and power is a coordinate-independent quantity.Because of this, we know that

VTb Fb = VT

a Fa. (3.94)

From Proposition 3.22 we know that Va = [AdTab ]Vb, and therefore Equa-tion (3.94) can be rewritten as

VTb Fb = ([AdTab ]Vb)TFa

= VTb [AdTab ]

TFa.

May 2017 preprint of Modern Robotics, Lynch and Park, Cambridge U. Press, 2017. http://modernrobotics.org


Wrench Representations in Different Frames

• The power generated by an (F ,V) pair must be the same regardless of theframe in which it is represented.

• Consider two frames {a} and {c}. We must have

VTc Fc = VTa Fa = ([AdTac]Vc)T Fa = VTc ([AdTac

])T Fa

• Since the above relation should hold for all possible twist Vc, we must have

Fc = [AdTac]T Fa

• We are often interested in fixed space frame {s} and body frame {b}, we candefine a spatial wrench Fs and body wrench Fb. They are related by

Fb = [AdTsb]T Fs


Example of Wrench

The robot hand is holding an apple with a mass of 0.1kg in a gravitational field g = 10m/s2

(rounded to keep the numbers simple) acting downward on the page. The mass of the hand is

0.5kg. What is the force and torque measured by the six-axis forcetorque sensor between the

hand and the robot arm?

110 3.4. Wrenches

yf

xf

za

xa

yh

xh

L2L1

g

Figure 3.22: A robot hand holding an apple subject to gravity.

Since this must hold for all Vb, this simplifies to

Fb = [AdTab ]TFa. (3.95)

Similarly,Fa = [AdTba ]TFb. (3.96)

Proposition 3.27. Given a wrench F, represented in {a} as Fa and in {b} asFb, the two representations are related by

Fb = AdTTab

(Fa) = [AdTab ]TFa, (3.97)

Fa = AdTTba

(Fb) = [AdTba ]TFb. (3.98)

Since we usually have a fixed space frame {s} and a body frame {b}, we candefine a spatial wrench Fs and a body wrench Fb.

Example 3.28. The robot hand in Figure 3.22 is holding an apple with a massof 0.1 kg in a gravitational field g = 10 m/s2 (rounded to keep the numberssimple) acting downward on the page. The mass of the hand is 0.5 kg. What isthe force and torque measured by the six-axis force–torque sensor between thehand and the robot arm?

We define frames {f} at the force–torque sensor, {h} at the center of massof the hand, and {a} at the center of mass of the apple. According to thecoordinate axes in Figure 3.22, the gravitational wrench on the hand in {h} isgiven by the column vector

Fh = (0, 0, 0, 0,−5 N, 0)

and the gravitational wrench on the apple in {a} is

Fa = (0, 0, 0, 0, 0, 1 N).



Statics of Open Chains

• Now consider an open-chain robot with n joints. Let τ ∈ Rn be the jointtorques vector.

• Applying torques to joints will result in motion of the robot and forces of theend effector. By conservation of power:

Power at the joints=(Power to move the robot)+(Power at the end-effector)

• At static equilibrium (i.e. no power is used to move the robot), we have

τT θ = FTb Vb = FTb Jb(θ)θ

• We can pick θ infinitesimally small, but in arbitrary direction in Rn.

⇒ τ = JTb (θ)Fb

• If we use the fixed space frame, we will have τ = JTs (θ)Fs

Statics of Open Chains Lecture 10 (ECE5463 Sp18) Wei Zhang(OSU) 9 / 12

End-Effector Force Analysis

• If an external wrench F is applied to the end-effector, the joint torques thatcan generate opposing wrench −F is given by

τ = JT (θ)(−F)

• What is the end-effector wrench generated by a given joint torque vector τ?

- the answer is(JT (θ)

)−1τ provided JT (θ) is invertible

- If JT (θ) is not invertible, the problem is not well defined.

- An interesting case is when JT (θ) has a nontrivial null space:

Null(JT (θ)) = {F ∈ R6 : JT (θ)F = 0}

- The wrench that lies in the null space causes no torques, i.e., the balanceequation is satisfied with τ = 0; the resisting forces are supplied completely bythe robot’s mechanical structure.


Example of Statics of Open Chains

What are the wrenches that can be resisted by the manipulator with τ = 0?

Chapter 5. Velocity Kinematics and Statics 207

L

θ1

θ2

L L

θ3

xb

yb

zbzs

xs

ys{s} {b}

Figure 5.20: RPR robot.

Exercise 5.8 The RPR robot of Figure 5.20 is shown in its zero position. Thefixed and end-effector frames are respectively denoted {s} and {b}.

(a) Find the space Jacobian Js(θ) for arbitrary configurations θ ∈ R3.(b) Assume the manipulator is in its zero position. Suppose that an external

force f ∈ R3 is applied to the {b} frame origin. Find all the directions inwhich f can be resisted by the manipulator with τ = 0.

Exercise 5.9 Find the kinematic singularities of the 3R wrist given the forwardkinematics

R = e[ω1]θ1e[ω2]θ2e[ω3]θ3 ,

where ω1 = (0, 0, 1), ω2 = (1/√

2, 0, 1/√

2), and ω3 = (1, 0, 0).

Exercise 5.10 In this exercise, for an n-link open chain we derive the analyticJacobian corresponding to the exponential coordinates on SO(3).

(a) Given an n × n matrix A(t) parametrized by t that is also differentiablewith respect to t, its exponential X(t) = eA(t) is then an n × n matrixthat is always nonsingular. Prove the following:

X−1X =

∫ 1

0

e−A(t)sA(t)eA(t)sds,

XX−1 =

∫ 1

0

eA(t)sA(t)e−A(t)sds.



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