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Higher-Order Kinematics in Dual Lie Algebra

Written By

Daniel Condurache

Submitted: September 28th, 2019 Reviewed: February 16th, 2020 Published: March 21st, 2020

DOI: 10.5772/intechopen.91779

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In this chapter, using the ring properties of dual number algebra, vector and tensor calculus, a computing method for the higher-order acceleration vector field properties in general rigid body motion is proposed. The higher-order acceleration field of a rigid body in a general motion is uniquely determined by higher-order time derivative of a dual twist. For the relative kinematics of rigid body motion, equations that allow the determination of the higher-order acceleration vector field are given, using an exponential Brockett-like formula in the dual Lie algebra. In particular cases, the properties for velocity, acceleration, jerk, and jounce fields are given. This approach uses the isomorphism between the Lie algebra of the rigid displacements se(3), of the Special Euclidean group, SE3,and the Lie algebra of dual vectors. The results are coordinate free and in a closed form.


  • higher-order kinematics
  • dual algebra
  • lie group

1. Introduction

The kinematic analysis of multibody systems has been traditionally considered as the determination of the positions, velocities, accelerations, jerks and jounces of their constitutive members. This is an old field with a long history, which has attracted the attention of mathematicians and engineers. Michel Chasles discovered (1834) that any rigid body displacement is equivalent to a screw displacement [1]. Screw theory is an efficient mathematical tool for the study of spatial kinematics. The pioneering work of Ball [2], the treatises of Hunt [3], and Phillips [4] and the multitude of contributions appearing in the literature are evidence of this. The isomorphism between screw theory and the Lie algebra, se(3), of the Special Euclidean group, SE3, provide with a wealth of results and techniques from modern differential geometry and Lie group theory [5, 6, 7, 8, 9].

A kinematic mapping relates the motion of a rigid body to the joint motions of a kinematic chain. Its time derivatives yield the twist, acceleration, jerk and jounce etc. of the body. Time derivatives of the twists of members in a kinematic chain and derivatives of screws are essential operations in kinematics. Recognizing the Lie group nature of rigid body motions, and correspondingly the Lie algebra nature of screws, Karger [5], Rico et al. [6], Lerbet [7] and Müller [8, 9] derived closed form expressions of higher-order time derivatives of twist.

In this chapter, using the tensor calculus and the dual numbers algebra, a new computing method for studying the higher-order accelerations field properties is proposed in the case of the general rigid body motion. For the spatial kinematic chains, equations that allow the determination of the nth order accelerations field are given, using a Brockett-like formula. The crucial observation is that the nth order time derivative of twist of the terminal body in a kinematic chain can be determined by propagating the kth order time derivative of twists of the bodies in the chain, for k=0,n¯. The results are coordinate-free and in a closed form.


2. Theoretical consideration on rigid body motion

The general framework of this chapter is a rigid body that moves with respect to a fixed reference frame R0. Consider another reference frame R originated in a point Q that moves together with the rigid body. Let ρQ denote the position vector of point Q with respect to frame R0, vQ its absolute velocity and aQ its absolute acceleration.

Then the vector parametric equation of motion is:


where ρ represents the absolute position of a generic point P of the rigid body with respect to R0 and R=Rt is an orthogonal proper tensorial function in SO3R. Vector r is constant and it represents the relative position vector of the arbitrary point P with respect to R.

The results of this section succinctly present the velocity and acceleration vector field in rigid body motion. These results lead to the generalization presented in the next section.

With the denotations that were introduced, the vector fields of velocities and accelerations are described by:




represent the velocity tensor respectively the acceleration tensor. Tensor Φ1=ωso3R is the skew-symmetric tensor associated to the instantaneous angular velocity ωV3R . Tensor Φ2=ω2+ε, where ε=ω̇ is the instantaneous angular acceleration of the rigid body. One may remark that vectors:


do not depend on the choice of point P of the rigid body. They are called the velocity invariant respectively the acceleration invariant (at a given moment of time).

2.1 The velocity field in rigid body motion

It is described by:


The instantaneous angular velocity ω of the rigid body may be determined as ω=vectΦ1. The major property that may be highlighted from Eq. (4) is that the velocity of a given point of the rigid may be computed when knowing the velocity tensor Φ1 and the velocity invariant a1:


2.2 The acceleration field in rigid body motion

It is described by


The absolute acceleration of a given point of the rigid body may be computed when knowing the acceleration tensor Φ2 and the acceleration invariant a2:


The instantaneous angular acceleration of the rigid body may be determined as:


The determinant of tensor Φ2 is (see [10]): detΦ2=ω×ε2. It follows that if ω×ε0, then tensor Φ2.

is invertible and its inverse is (see [10]):


It follows that if tensor Φ2 is non-singular, then for an arbitrary given acceleration a we may find a point of the rigid that has this acceleration. Its absolute position is given by (see also Eq. (8)):


Particularly, if Φ2 is non-singular, then there exists a point G of zero acceleration, named the acceleration center. Its absolute position vector is given by:


3. The vector field of the nth order accelerations

This section extends some of the previous considerations to the case of the nth order accelerations. We define the nth order acceleration of a point as:


For n=1, it represents the velocity, and for n=2, the acceleration. By derivation with respect to time successively in Eq. (2), it follows that:


We define:


the nthorder acceleration tensor in rigid body motion. A vector invariant is immediately highlighted from Eq. (14) with the denotation (15). Vector:


does not depend on the choice of the point of the rigid body for which the acceleration an is computed. Vector an is named the invariant vector of the nthorder accelerations. Then Eq. (7) may be generalized as it follows:


The next Theorem gives the fundamental properties of the vector field of the nth order accelerations.

Theorem 1. In the rigid body motion, at a moment of timet, there exist tensorΦndefined byEq. (15)and vectoransuch as:


for any point Pof the rigid body with the absolute position defined by vector ρ.

Remark 1. Given the absolute position of a point of the rigid body and knowingΦnandan, its acceleration is computed from:


Remark 2. TensorΦnand vectorangeneralize the notions of velocity/acceleration tensor respectively velocity/acceleration invariant. They are fundamental in the study of the vector field of thenthorder accelerations. The recursive formulas for computingΦnandanare:


Remark 3. One may remark that fromEq. (20)it follows by direct computation:


Remark 4. By defining thenthorder instantaneousnthorder angular acceleration of the rigid bodyεndn1dtn1ω, it follows fromEq. (21)that its associated skew-symmetric tensor may be expressed asεn=dn1dtn1Φ1. The expression of the instantaneousnthorder angular acceleration is:


3.1 Homogenous matrix approach to the field of nth order accelerations

The set of affine maps, g:V3V3,gu=Ru+w, where R is an orthogonal proper tensor and w a vector in V3 is a group under composition and it is called the group of direct affine isometries or rigid motions and it is denoted SE3. Any rigid finite motion may be described by such a map. Tensor R models the rotation of the considered rigid body and vector w its translation. An affine map from SE3 may be represented with a 4×4 square matrix:


One may remark that the following relations hold true:


We may extend now SE3 to SE3R, the set of the functions with the domain ℝ and the range SE3. The parametric vector equation of the rigid body motion (1) may be rewritten with the help of a homogenous matrix function in SE3R like it follows:


From Eq. (25), it follows that:


and by making the computations and taking into account Eqs. (3) and (4) it follows that:


By using the previous considerations, it follows that Eq. (25) may be extended like:


Eq. (28) represents a unified form of describing the vector field of the nth order accelerations in rigid body motion. The matrix:


contains both the nth order acceleration tensor Φn and the vector invariant an. Eqs. (20) may be put in a compact form:


If follows that Ψn may be written as:


4. Symbolic calculus of higher-order kinematics invariants

We will present a method for the symbolic calculation of higher-order kinematics invariants for rigid motion.

Let be an and Φn, nN vector invariant, respectively, tensor invariant for the nth order accelerations fields. We denote by


and we have the following relationship of recurrence:


The pair of vectors ωv is also known as the spatial twist of rigid body.

Let be A the matrix ring


and AX the set of polynomials with coefficients in the non-commutative ring A. A generic element of AX has the form


Theorem 2. There is a unique polynomialPnAXsuch thatΨnwill be written as


whereD=ddtis the operator of time derivative.

Proof: Taking into account Eqs. (36) and (33) we will have the following relationship of recurrence for PnD:


Since Ψ1=ωv00 it follows the next outcome.

Theorem 3. There is a unique polynomial with the coefficients in the non-commutative ringLV3V3such that the vector respectively the tensor invariants of thenthorder accelerations will be written as


wherePnfulfills the relationship of recurrence


It follows


Thus, it follows:

  • the velocity field invariants


  • the acceleration field invariants


  • jerk field invariants


  • hyper-jerk (jounce) field invariants


Remark 5. The higher-order time derivative of spatial twist solve completely the problem of determining the field of thenthorder acceleration of rigid motion.

4.1 Higher-order acceleration center and vector invariants of rigid body motion

Equation (16) may be written as


This shows us that the vector function


has the same value in every point of the rigid body under the general spatial motion, at a given moment of time t. It represents a vector invariant of the n-th order acceleration field.

The invariant value of vector In is obtained for ρ=0 and it is the n-th order acceleration of the point of the rigid body that passes the origin of the fixed reference frame at a given moment of time: In=a0nan. Eq. (46) becomes:


Let be Φn be the adjugate tensor of Φn uniquely defined by:ΦnΦn=detΦnI.

From Eq. (46), results another invariant


The value of this invariant is Jn=Φnan.

In the specific case when tensor Φn is non-singular (detΦn0), from (47) results the position vector having an imposed n-th order acceleration a:


In a particular case of the n-th order acceleration centerGn (i.e. the point that have a=0) on obtain:


Assuming that the tensor Φn is non-singular, the previous relations lead to a new vector invariant that characterize the accelerations of n-th and m-th order (n,mN):


The value of this invariant is Km,n=amΦmΦn1an.

The problem of the determination the adjugate tensor of the n-th acceleration tensor and the conditions in which these tensors are inversable is, as the author knows, still an open problem in theoretical kinematics field. We will propose a method based on the tensors algebra that will give a closed form, coordinate- free solution, dependent to the time derivative of spatial twist.

The vector field of the higher-order acceleration is a non-stationary vector field. Differential operator div and curl is expressed, taking into account Eq. (47), through the linear invariants of the tensor Φn, as below:


Let ΦLV3V3 a tensor and we note t=vectΦ and S=symΦ. The below theorem takes place.

Theorem 4. The adjugate tensor and determinant of the tensorΦis:


Let Φn the n-th order acceleration tensor, Φn=tn+Sn.

The vectors tn and the symmetric tensors Sn,nN can be obtained with the below recurrence relation:


It follows that:

  • Velocity field: Φ1=ω, t1=ω, S1=0


Φ1 is singular for any ω. In this case,


  • Acceleration field: Φ2=ω2+ω̇,t2=ω̇,S2=ω2


Φ2 is nonsingular if and only if ω×ω̇0. In this case


  • Jerk field: Φ3=ω¨+2ω̇ω+ωω̇+ω3,t3=ω¨+12ω̇×ωω2ω,S3=32ωω̇+ω̇ω,


Φ3 is nonsingular if and only if 4ω̇×t3ω×t39ω·ω̇ω×ω̇2. In this case


  • Jounce field:


In Eqs. (65) and (66), the following notation has been used:




then Φ4 is inversible and


In the hypothesis (68), there is jounce center, determined by


5. Dual algebra in rigid body kinematics

In this section, we will present some algebraic properties for dual numbers, dual vectors and dual tensors. More details can be found in [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

5.1 Dual numbers

Let the set of real dual numbers to be denoted by


where a=Rea¯ is the real part of a¯ and a0=Dua¯ the dual part. The sum and product between dual numbers generate a ring with zero divisors structure for R¯.

Any differentiable function f:SRR,f=fa can be completely defined on S¯R¯ such that:


Based on the previous property, two of the most important functions have the following expressions: cosa¯=cosaεa0sina;sina¯=sina+εa0cosa;

5.2 Dual vectors

In the Euclidean space, the linear space of free vectors with dimension 3 will be denoted by V3. The ensemble of dual vectors is defined as:


where a=Rea¯ is the real part of a¯ and a0=Dua¯ the dual part. For any three dual vectors a¯,b¯,c¯, the following notations will be used for the basic products: a¯·b¯scalar product, a¯×b¯cross product and a¯b¯c¯=a¯·b¯×c¯—triple scalar product. Regarding algebraic structure, V¯3+·R is a free R¯-module [13, 14].

The magnitude of a¯, denoted by a¯, is the dual number computed from


where . is the Euclidean norm. For any dual vector a¯V¯3, if a¯=1 then a¯ is called unit dual vector.

5.3 Dual tensors

An R¯-linear application of V¯3 into V¯3 is called an Euclidean dual tensor:


Let LV¯3V¯3 be the set of dual tensors, then any dual tensor T¯LV¯3V¯3 can be decomposed as T¯=T+εT0, where T,T0LV3V3 are real tensors. Also, the dual transposed tensor, denoted by T¯T, is defined by


while v¯1,v¯2,v¯3V¯3, Rev¯1v¯2v¯30 the determinant is


Orthogonal dual tensor maps are a powerful instrument in the study of the rigid motion with respect to an inertial and non-inertial reference frames.

Let the orthogonal dual tensor set be denoted by:


where SO3¯ is the set of special orthogonal dual tensors and I¯ is the unit orthogonal dual tensor.

Theorem 5 (Structure Theorem). For anyR¯SO¯3a unique decomposition is viable


whereRSO3andρV3are called structural invariants.

Taking into account the Lie group structure of SO¯3 and the result presented in previous theorem, it can be concluded that any orthogonal dual tensor R¯SO¯3 can be used globally parameterize displacements of rigid bodies.

Theorem 6 (Representation Theorem). For any orthogonal dual tensorR¯defined as inEq. (79), a dual numberα¯=α+εdand a dual unit vectoru¯=u+εu0can be computed to have the following equation, [13, 14, 15]:


The parameters α¯ and u¯ are called the natural invariants of R¯. The unit dual vector u¯ gives the Plücker representation of the Mozzi-Chasles axis [13, 14], while the dual angle α¯=α+εd contains the rotation angle α and the translated distance d.

The Lie algebra of the Lie group SO¯3 is the skew-symmetric dual tensor set denoted by so¯3=α¯LV¯3V¯3α¯=α¯T, where the internal mapping is α¯1α¯2=α¯1α¯2.

The link between the Lie algebra so¯3, the Lie group SO¯3, and the exponential map is given by the following.

Theorem 7. The mapping


is well defined and onto.


and is the inverse of Eq. (81).

Based on Theorems 6 and 7, for any orthogonal dual tensor R¯, a dual vector ψ¯=α¯u¯=ψ+εψ0 can be computed and represents the screw dual vector, which embeds the screw axis and screw parameters.

The form of ψ¯ implies that ψ¯logR¯. The types of rigid displacements that can be parameterized by ψ¯ are:

  • roto-translation if ψ0,ψ00andψ·ψ00ifψ¯R¯ andψ¯εR;

  • pure translation if ifψ=0andψ00ifψ¯εR¯;

  • pure rotation if ψ0andψ·ψ0=0ifψ¯R.

Theorem 8. The natural invariantsα¯=α+εd,u¯=u+εu0can be used to directly recover the structural invariantsRandρfromEq. (79):


Theorem 9 (isomorphism theorem). The special Euclidean groupSE3·andSO¯3·are connected via the isomorphism of the Lie groups


whereg=Rρ01, RSO¯3, ρV3.

Remark 6. The inverse ofΦ is


whereR=ReR¯,ρ=vectDuR¯·RT .


6. Higher-order kinematics in dual Lie algebra

Being the rigid body motion given by the following parametric equation in a given reference frame:


with tIR is time variable.

The dual orthogonal tensor that describes the rigid body motion is [13, 24]:


In relation (87), the skew symmetric tensor associated to the vector ρ is denoted by ρ.

It can be easily demonstrated [14, 15], that:


The tensor R transports the dual vectors from the body frame in the space frame with the conservation of the dual angles and the relative orientation of lines that corresponds to the dual vectors a¯andb¯.

The dual angular velocity for the rigid body motion (86) is given by (87):


It can be demonstrated that:




is the instantaneous angular velocity of the rigid body and


is the linear velocity of a point of the rigid body that coincides with the origin of the reference frame at that given moment.

The dual angular velocity ω¯ completely characterizes the distribution of the velocity field of the rigid body. The pair (ω,v) is called “the twist of the rigid body motion” [13, 14].



the dual velocity for a point localized in the reference frame by the position vector ρ.

In (93), ω is the instantaneous angular velocity of the rigid body and vρ is the linear velocity of the point. Using the next equation,


from (90), (92)(94), results:


Consequently, eεrV¯r is an invariant having the same value for any r.

Writing this invariant in two different points of the rigid body (noted with P and Q ), results that:


From (97), results:


with PQ=ρQρP.

Relation (97) is true for any P and Q.

Analogue with Eq. (95), the following invariants take place:


where we denoted


with Aρ,Jρ,Hρ the reduced acceleration, reduced jerk, respectively the reduced hyper-jerk (jounce), in a point given by the position vector ρ:


In (100), aρ, jρ and hρ are, respectively, the acceleration, the jerk, and the hyper-jerk (jounce), in a point given by the position vector ρ.

Analogue with Eq. (97) the following equations take place:


The lines corresponding to the dual vectors ω¯̇,ω¯¨,ω¯... represent the loci, where the vectors Aρ,Jρ,Hρ have the minimum module value. Supplementary,


Interesting is the fact that for the plane motion minAρ=minJρ=minHρ=0 because Duω¯̇=Duω¯¨=Duω¯...=0

All properties are extended for higher-order accelerations. The vector ω¯n=dnω¯dtn,nN describes completely the helicoidally field of the n order reduced accelerations, for nN:


In Eq. (103)A¯ρn denote the nth order of the dual reduced acceleration in a given point by the position vector ρ.

It follows that the dual part of the nth order differentiation of ω¯n


is the nth order reduced acceleration of that point of the rigid body that at the given time pass by the origin of the reference frame.

From equation


it follows that


with the following notations


for the nN order acceleration of the point given by the position vector ρ and


for the nth order reduced acceleration of the same point the equation:


which proves the character of the helicoidally field of the nth order reduced accelerations field.

For ρ=0, the relations between the nth order reduced acceleration and the n order acceleration from point O, the origin of the reference frame, are written


The invert of previous equation is written:


where Pn is the polynomial with the coefficients in the ring of the second order Euclidean tensors and the polynomials PnD follow the recurrence equation:


it follows successively


If we denote T=vω and by Ψn=anΦn, nN, for the case of the velocities, accelerations, jerks and jounces, on obtain (Figure 1):

Figure 1.

Higher-order time derivative of dual twist.


Theorem 10. Thenthorder accelerations field of a rigid body in a general motion is uniquely determined by thekthorder time derivative of a dual twistω¯, k=0,n1¯.


7. Higher-order kinematics of spatial chain using dual Lie algebra

Consider a spatial kinematic chain of the bodies Ck,k=0,m¯ where the relative motion of the rigid body Ck with respect to Ck1 is given by the proper orthogonal tensor R¯k1kSO¯3R. The relative motion properties of the body Cm with respect to C0 are described by the orthogonal dual tensor (Figure 2):

Figure 2.

Orthogonal dual tensors of relative rigid body motion.


Instantaneous dual angular velocity (dual twist) of the rigid body in relation to the reference frame it will be given by the equation


It follows from (110) and (111) that:




Using the denotation


Eq. (118) will be written


where ω¯k is the dual twist of the relative motion of the body Ck in relation to the body Ck1 observed from the body C0.

Remark 7. Form=2,ω¯20=ω¯1+ω¯2, we will obtain the space replica of Aronhold-Kennedy Theorem: the instantaneous screw axis for the three relative rigid body motions has in every moment a common perpendicular, at any given time. The common perpendicular is line that corresponds to the dual vectorω¯1×ω¯2.

To determine the field of the nth order accelerations of a rigid body Cm we have to determine the ω¯mn0, nN.

We denote ω¯pn=R¯10R¯21R¯p1p2Ω¯pn the nth order derivative of the relative dual twist Ω¯p, resolved in the body frame of C0.

In order to determine the nth order accelerations field of a rigid body Cm, we have to determine the ω¯mn0, nN.

To compute ω¯0mn,nN we will use the following

Lemma: Ifω¯p=R¯Ω¯pwithR¯SO¯3Randω¯p,Ω¯pV¯3R, then


wherepnω¯are polynomials of the differential operatorD=ddt, with coefficients in the non-commutative ring of Euclidian dual tensors.


whereCnkis the binomial coefficient, Dkωp=ωpkandΦ¯pare dual tensors


which follow the recurrence equation:


Theorem 11. The following equation takes place


wherepnωare polynomials of the derivative operatorD=ddt, with coefficients in the non-commutative ring of Euclidian dual tensors


whereCnkis the binomial coefficient, Dkω¯p=ω¯pk and Φ¯pare dual tensors


which follow the recurrence equation:


Other equivalent forms of Eq. (127) are the following recursive formulas (Figures 3 and 4):

Figure 3.

Higher-order time derivative of dual twist of relative motion.

Figure 4.

Higher-order time derivative of dual twist of relative motion on terminal body.


The previous equations are valid in the most general situation where there are no kinematic links between the rigid bodies C1,C2,,Cm.

The following identity can be proved:


where Cnk1,,kp1=n!k1!kp1! is the multinomial coefficient.

From Eq. (131), on obtain the closed form non-recursive coordinate-free formula:





8. Higher-order kinematics for general 2C manipulator

We’ll apply the general results obtained in the previous chapter for the particular case of four degrees of freedom 2C general manipulator. In this case the relative motions of three bodies C0,C1,C2 are given, the spatial motion of the terminal body C2 been described by dual orthogonal tensor as it follows:




In Eqs. (138) and (139), the dual angles α¯1t and α¯2t are four times differentiable functions, and unit dual vectors u¯10 and u¯21 being constant. To simplify the writing, we will denote:


According to the observations from Section 6, the vector field of the velocity, the acceleration, the jerk, the jounce is uniquely determined by the dual vectors ω¯,ω̇¯,ω¨¯,ω...¯. Taking into account Eq. (133), we will have:


Similarly, the results for six degrees of freedom general 3 C manipulator can be obtained, the calculus being a little longer.


9. Conclusions

The higher-order kinematics properties of rigid body in general motion had been deeply studied. Using the isomorphism between the Lie group of the rigid displacements SE3 and the Lie group of the orthogonal dual tensors SO¯3, a general method for the study of the field of arbitrary higher-order accelerations is described. It is proved that all information regarding the properties of the distribution of high-order accelerations are contained in the n-th order derivatives of the dual twist of the rigid body. These derivatives belong to the Lie algebra associated to the Lie group SO¯3.

For the case of the spatial relative kinematics, equations that allow the determination of the n-th order field accelerations are given, using a Brockett-like formulas specific to the dual algebra. In particular cases the properties for velocity, acceleration, jerk, hyper-jerk (jounce) fields are given.

The obtained results interest the theoretical kinematics, jerk and jounce analysis in the case of parallel manipulations, control theory and multibody kinematics.


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Written By

Daniel Condurache

Submitted: September 28th, 2019 Reviewed: February 16th, 2020 Published: March 21st, 2020