Explicit Lagrangian Formulation of the Dynamic Regressors for Serial Manipulators

Explicit Lagrangian Formulation of the Dynamic Regressors for Serial Manipulators M Gabiccini 1 A Bracci 1 D De Carli M Fredianelli A Bicchi 1 Dipartimento di Ingegneria Meccanica Nucleare e della Produzione Università di Pisa Italy E-mail: {mgabiccini andreabracci@ingunipiit Centro Interdipartimentale di Ricerca E Piaggio Università di Pisa Italy E-mail: {ddecarli mfredianelli bicchi@ingunipiit Keywords: serial manipulators dynamic regressors Lagrangian formulation In this paper an explicit formulation of the matrices used in the linear-regressor form of the dynamics of general n-dof serial manipulators is presented First a closed formula for the classical regressor Y q q q) is derived directly from the Lagrange equations in terms of the Denavit-Hartenberg acobians the configuration vector q and its derivatives q q hen a specialized version Y r q q q r q r ) is obtained which is suitable for application of the Slotine-Li adaptive control scheme he advantage of the proposed formulation is that an explicit analytic form of the regressor of a general manipulator can be produced systematically a dedicated Mathematica M package is made available) Furthermore the analytic form of the regressor enables direct inspection of the physical properties of the manipulator which is concealed by the recursive formulations available in the literature o illustrate the proposed approach explicit calculations are carried out for a simple two-dof manipulator 1 INRODUCION he manipulator regressor usually denoted as Y q q q) is a matrix function originated by Atkeson 1 Khosla and Kawasaki 3 employed to write the manipulator dynamics linearly in the inertial parameters Bq) q + Cq q) q + Gq) Y q q q)π τ 1) where q q and q R n denote the joint position velocity and acceleration respectively; Bq) Cq q) R n n are the inertia and Coriolis matrices; Gq) R n is the gravitational force vector he left side of 1) represents the inertial and the gravitational forces in the dynamics of a n-link rigid-body manipulator while the right side is the input torque/force vector τ R n As evident from 1) the robot dynamic equations are linear with respect to a properly defined inertia parameter vector π R r via the regressor matrix Y q q q) R n r he manipulator regressor Y q q q) is a key quantity in derivation as well as implementation of many established adaptive motion and force control algorithms 4 A different definition of the manipulator regressor stems from the Slotine-Li adaptive algorithm 5 where a novel Y r q q q r q r ) is introduced to fit Bq) q r + Ĉq q) q r + Gq) Y r q q q r q r )π ) where q r q d Λq q d ) is the reference velocity; Λ R n n is an arbitrary pd matrix and q d is the desired joint trajectory It is worth remarking here that in 1) multiple forms of Cq q) can

be employed as far as Cq q) q is the same On the other hand for the Slotine-Li adaptive control scheme to work properly it is required for the regressor to be written in a form Ĉq q) obtained eg via the Christoffel symbols which ensures the skew-symmetry of matrix Ḃ Ĉ In principle the regressor can be obtained by following a two-step indirect approach as: i) one first formulates the manipulator dynamics in the standard form 1 Bq) q + Cq q) q + Gq) τ 3) and ii) having defined a vector of parameters π one works on every entry on the left-hand side of 3) to obtain the desired version Y π τ Even if the first step is straightforward the parametric extraction is hardly systematic and computationally demanding as the entries of π are in general spread all over the entries of Bq) Cq q) and Gq) Moreover for manipulators with many degrees of freedom the computational burden can be hindering even for most advanced computer algebra systems eg Mathematica M 6 ) Motivated by the necessity of finding a direct way to obtain the regressor dynamics several solutions have been proposed in the literature Atkeson et al 1 proposed a recursive Newton-Euler approach but did not address the Slotine-Li version of the regressor Gautier and Khalil 7 as well as Mayeda et al 8 focused mainly on the identification of the minimum set of parameters actually appearing in the equations of motion he work of Lu and Meng 9 uses the Lagrangian approach to derive a closed formula for the classical regressor heir analytical treatment does not generalize to the computation of the Slotine- Li regressor and is quite different from the one developed in this paper Following the success of the Slotine-Li adaptive control scheme 5 several researchers have studied a regressor formulation that is compatible with the specific requirements of the method ie the derivation of those forms Ĉq q) of the Coriolis matrix that ensures skew-symmetry of Ḃ Ĉ An approximate algorithm based on a Newton-Euler recursive formulation is provided already in 5 Yuan and Yuan 10 employ a modified version of the Newton-Euler equations proposed in 11 to inject the reference velocities and accelerations in the expressions he recursive nature of the algorithm entails that a formula for the regressor matrix Y r q q q r q r ) is not obtained rather values are provided for the matrix product used in Slotine-Li parameter update equation ie Y s) In this work a different approach is pursued By employing the Lagrange equations of motion a closed formula for the classical regressor Y q q q) is obtained by properly factorizing the expressions in the inertial parameters throughout the computations Moreover by directly imposing skew-symmetry of the terms corresponding to N Ḃ C and by introducing the reference velocity the closed formula for the Slotine-Li regressor Y r q q q r q r ) alone is derived he advantage of our formulation is that an explicit analytic form of the regressor of a general manipulator can be produced in a thoroughly systematic way Furthermore the exact form of the regressor we provide enables analytic inspection of the physical properties of the manipulator such as eg the possible irrelevance of some of the inertial parameters corresponding to a zero column in Y )) which is concealed by the recursive formulations available in the literature he algorithm is implemented in a Mathematica M package which is made available to the public o illustrate the proposed approach explicit calculations are carried out for the simple two-dof planar elbow manipulator 1 he indirect route to the definition of the Slotine-Li regressor Y r q q q r q r ) would entail the use of ) and is here omitted for brevity

LAGRANGIAN EQUAIONS OF ROBO DYNAMICS o establish the background and notation for the rest of the paper we briefly review the Lagrangian formulation of the dynamics of a general n-dof serial manipulator Let U i) and i) be respectively the potential and kinetic energy associated to the link i and q the n 1 vector of joint displacements; the Lagrangian of the serial chain is defined as L q q) q q) Uq) i1 ) i) q q) U i) q) because L q q) is obviously linkwise additive hus adopting the Lagrange equations the dynamics of the manipulator is d L dt q L i1 d L i) dt q L i) q q) 4) i1 Li) τ 5) where τ is the n 1 vector of applied joint torques or forces he key feature of robot dynamics exploited to derive adaptive control laws 4 5) is its linearity with respect to some properly defined parameters π Y q q q) π τ 6) where Y q q q) is the manipulator regressor Referring to 5) and denoting the block of the regressor associated to the link i with we have i1 d L i) dt q Li) 7) where R ri is the vector of the parameters associated to link i and R n ri hus for a n-link manipulator we can write the complete regressor as Y Y 1) Y n) R n r and the dynamic parameters as π π 1) π n) R r with r n ii r i Considering 4) and 7) it follows that d i) i) + dt q i1 i) U 8) 3 DIREC FORMULAION OF HE MANIPULAOR REGRESSOR With reference to Fig 1 we assume that each link i has a local coordinate system {i with the origin O i fixed at joint i + 1 according to the classical Denavit-Hartenberg DH) conventions A coordinate system is fixed at the center of mass G i located with respect to O i by the vector p igi and with respect to the global origin O by 0 p igi 31 Kinetic Energy erms Applying the König theorem with respect to the global coordinate system O the kinetic energy i) of the i-th link can be written as i) 1 m i 0 vg 0 i v Gi + 1 i i I i Gi ω i 9)

where 0 v Gi is the G i linear velocity i ω i is the angular velocity and i I Gi is the moment-of-inertia tensor about the center of mass G i Furthermore using the rotation matrix 0 R i from the global frame {0 to the D-H frame {i we obtain i ω i i R 0 0 ω i 0 R i 0 ω i 10) Next adopting the position and orientation DH acobians vi and we have 0 v Gi 0 v i + 0 ω i 0 p igi vi q + q 0 R i p igi ; 0 ω i q 11) hus substituting 10) and 11) in 9) we obtain i) 1 m i vi q + q 0 R i p igi ) vi q + q 0 R i p igi ) + 1 q ω i 0 R i i I Gi 0 R i ) q Let Sx) R 3 3 be a skew-symmetric matrix such that Sx) y S y) x x y for any x y R 3 ; we have SR x) R Sx) R where R R 3 3 represents a rotation matrix Considering this latter property we can write i) 1 m i q v i vi ) q 1 m i q { v i S 0 R i p igi ) q + 1 m i q { ω i S 0 R i p igi ) vi q 1) + 1 q { ω i 0 R i i I Gi + m i S p igi ) Sp igi ) 0 R i q joint axis i-1 joint axis i joint axis i+1 link i-1 v G i x G i ω i z G i G i y G i link i z i-1 0p ig i p ig i y i z i O x i i O i-1 y i-1 x i-1 z 0 0p i y O 0 x 0 Figure 1: Local coordinate systems and vectors for the i-th link

aking the derivative of 1) according to 8) we get i) q m i q v i vi ) m i q { v i S 0 R i p igi ) + mi q { ω i S 0 R i p igi ) vi + q { ω i 0 R i i I i 0 R i 13) where i I i i I Gi + m i S p igi ) Sp igi ) is the inertia tensor with respect to origin O i Steiner s theorem) Since i I i is a symmetric matrix and thanks to the properties of skew-symmetric matrices we can rearrange 13) as i) q v i vi ) q m i + { v i S q) 0 R i ω i S vi q) 0 R i mi p igi + ω i 0 R i i I i 0 R i q 14) Eq 14) is made of three terms: the first and the second ones are obtained by extracting respectively the mass m i and the first order moment-of-inertia m i p igi ; the elements of the matrix i I i have to be extracted from the third term For such purpose we can consider a third-order tensor E R 3 3 6 and the vector of parameters i ixx ixy ixz iyy iyz izz where E E 1 E E 3 E 4 E 5 E 6 i I i ixx ixy ixz ixy iyy iyz ixz iyz izz 15) with E 1 E 4 1 0 0 0 0 0 E 0 0 0 0 0 0 0 1 0 E 5 0 0 0 0 1 0 1 0 0 E 3 0 0 0 0 0 0 0 0 1 E 6 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 In this way we can write the tensor i I i as the inner product between E and i Hence With this procedure the third term of 14) becomes i I i E i 16) ω i 0 R i i I i 0 R i q ω i 0 R i E 0 R i q i ω i 0 R i E 1 0 R i q ω i 0 R i E 6 0 R i q i 17) Referring to 8) we are now able to extract the mass m i and both m i p igi and ilm l m x y z) from the first term of the Lagrange equations hus we get d dt i) q Ẋi) 0 0 + Ẋi) 1 1 + Ẋi) 18)

where X i) 0 v i vi ) q R n 1 X i) 1 { v i S q) ω i S vi q) 0 R i R n 3 X i) ω i 0 R i E1 E E 6 0 R i q R n 6 0 m i R 19) 1 m i p igix m i p igiy m i p igiz R 3 0) i R 6 1) he second term of the Lagrange equations 8) is a function of the kinetic energy as well With reference to 1) we differentiate with respect to q Hence i) 1 + 1 + 1 { q v i vi ) q m i { q vi S q) 0 R i ω i S vi q) 0 ) R i m i p igi { q 0 R i i I 0 i R ) i q Let us consider the third term of the right part of the previous equation; again by employing 16) we can extract the second order moment-of-inertia i It follows that { 0 R i i I 0 i R 0 R i E 0 1 R ) i i i 0 R i E 0 6 R ) i hus we can now write more compactly ) where i) W i) 0 1 q W i) 1 1 W i) 1 q W i) 0 0 + W i) 1 1 + W i) 3) ) 1 vi vi q ) n vi vi 0 1 R i S q) vi q 0 R i S vi q) q 0 n R i S q) vi q 0 R i S vi q) q 1 0 R i E 0 R ) i q n 0 R i E 0 R ) i 4)

3 Potential Energy erm he last term of the Lagrange equations to be managed comes from the potential energy he potential energy U i) associated to the link i can be written as U i) m i g 0 p i + 0 R i p igi ) 5) where g is the gravitational acceleration vector with respect to the global frame and 0 p i is the position vector from the origin O of the global coordinate system to the DH frame {i fixed at O i We can derive the expression used in the Lagrange equations differentiating U i) with respect to q Hence U i) m i { g 0 p i + g 0 R i g p igi v 0 R i ) i g m i m i p igi 6) Considering the last term of the right part of the previous equation we get g 0 R i m i p igi ) { m i p igi ) 0 R i g ) 0 R i g) 1 ) 0 R i g) m i p igi In this way we can express the term associated with potential energy in the Lagrange equations as a sum of two linear functions with respect to m i and m i p igi Equation 6) can also be written more compactly as U i) Z i) 0 m i + Z i) 1 m i p igi 7) where Z i) 0 v i g Z i) 1 ) 0 R i g) 1 ) 0 R i g) n n 8) 33 Regressor and Inertial Parameters he Lagrange equations related to the link i can now be written from 8) 18) 19) 0) 1) 3) and 7) as where d i) i) + dt q i) U 9) 0 1 0 1 30)

with 0 Ẋ i) 0 W i) 0 + Z i) 0 R n 1 31) 1 Ẋ i) 1 W i) 1 + Z i) 1 R n 3 3) Ẋ i) W i) R n 6 33) herefore by simply juxtaposing the regressor blocks associated to the link i for i 1 n the manipulator regressor can be written as Y q q q) Y 1) Y n) 34) Similarly we build the inertial parameters vector by stacking the for each link π π 1) π n) 35) 4 DIREC FORMULAION OF HE SLOINE-LI REGRESSOR he result of the previous section ie the direct formulation of the classical dynamic regressor lends itself to generalization to a form that is compatible with the Slotine-Li version of the regressor Indeed the Slotine-Li SL) control algorithm and adaptation law are obtained introducing a robot regressor defined as 5 Bq) q r + Ĉq q) q r + Gq) Y r q q q r q r ) π 36) where q r was defined in ) and Y r q q q r q r ) is denoted as the SL regressor It is worth noting the fundamental role played in 36) by that particular choice of Ĉq q) since Slotine and Li 5 proved the stability of their adaptive controller using the skew-symmetric identity s Ḃq q) Ĉq q)s 0 s 0 Rn 37) In order to satisfy 37) matrix Ĉq q) can be built via the Christoffel symbols of the first kind 11 With this choice the hj-th element of Ĉq q) can be written as ĉ hj 1 bhj + b hk b ) jk q k 1 ĉ hjk q k 38) k j h k1 where b hj is the hj-th element of Bq) Let us now present an alternative form of 5) which is useful for further developments d dt q B q + Ḃ 1 B q q 39) where B Rn n n B is a third-order tensor with elements b jk h Here Cq q) Ḃq) 1 q Bq) Recalling the identity Cq q) q is the classical Coriolis matrix parentheses) eq39) can be written in component-wise fashion as Bq) q + Ĉq q) q h b hj q j + j1 k1 jkh Ĉq q) q note the same velocity q is inside and outside j1 k1 1 bhj k + b hk j ) q k b jk q k q j 40) h

In this form the injection of the reference velocity and acceleration q r and q r as required from 36) leads to Bq) q r + Ĉq q) q r h n 1 bhj b hj q rj + + b ) hk q k b jk q k q rj k j h j1 j1 k1 A further useful step to take which will be clearer later) is to operate a different aggregation of the summation terms in 41) by defining X and W whose h-th components X h and W h are explicitly given by X h b hj q rj + j1 j1 k1 1 bhj + b ) hk q k q rj W k h 1 j j1 k1 41) b jk h q k q rj 4) By handling with care the tensor dimensions the terms in 4) can be cast in matrix notation as X Bq) q r + 1 {B + B ) 13 q qr W 1 31 q qr 43) {B Here the generalized transpose operator was employed which is defined for tensors of arbitrary order If we let A) i1 i j i k a i1 i ji k be a tensor of order k the operator ) n 1 n j n k allows to specify an arbitrary reordering to apply to the indices of a tensor such that if à A n 1 n j n k the j-th level in à corresponds to the n j-th level in A as follows à ) i 1 i j i k a in1 i nj i nk 44) It is also worth pointing out that X and W are linkwise addiditive that is X X i) W i1 W i) 45) i1 since Bq) itself enjoys the same property In fact with reference to 1) and 13) this can be written as Bq) Bq) i) i1 Bq) i) m i v i vi ) + m i { S 0 R i p igi ) vi v i S 0 R i p igi ) + { ω i 0 R i i I Gi + m i S p igi ) Sp igi ) 0 R i Now meaningful expressions of 4) and/or 43) consistent with definition 46) and explicitly linear in the dynamic parameters can be sought for he first term in 46) is already linear in the i-th link mass m i ; in order to extract the first order moment-of-inertia the second term in the right-hand side of 46) can be rearranged as 46) m i { 0 R i Sp igi ) 0 R i vi v i 0 R i Sp igi ) 0 R i 47)

hen as done for 16) Sp igi ) is reshaped as an inner product of the third order tensor Q R 3 3 3 with vector p igi p igix p igiy p igiz thus getting Sp igi ) Q p igi where Q Q 1 Q Q 3 Q 1 0 0 0 0 0 1 Q 0 1 0 0 0 1 0 0 0 Q 3 1 0 0 0 1 0 1 0 0 48) 0 0 0 Furthermore by following the same steps as in 17) 46) is rewritten as Bq) i) m i v i vi ) + { ω i 0 R i Q 0 R i vi v i 0 R i Q 0 R i mi p igi + { ω i 0 R i E 0 R i i 49) By substituting 49) in each term of 45) and eliciting the structure of each X i) and W i) from the global versions 43) we are in a position to write 41) linearly in the parameters π edious but systematic calculations lead us to the explicit form X i) Ẋi) 0 + Ẋi) 1 + Ẋi) r 50) where { Ẋ i) v i vi ) q r + 1 vi vi ) Ẋ i) X i) 1 q r + 1 Ẋ i) r X i) q r + 1 Xi) 1 Xi) + vi vi ) q q r ) X i) 134 1 + q q r 51) ) X i) 134 + q q r ) 13 with X i) 1 ω i 0 R i Q1 Q Q 3 0 R i vi v i 0 R i Q1 Q Q 3 0 R i R n n 3 X i) ω i 0 R i E1 E E 6 0 R i R n n 6 5) In 51) the generalized transpose operator was applied to both 3-rd and 4-th order tensors Representatives of these classes appearing in 51) are respectively v i vi ) R n n n X i) R n n 6 n 53) he terms originating from W in 4) can actually be computed as for Y q q q) with minor differences hese result in W i) W i) 0 + W i) 1 + W i) r 54)

where W i) 1 q r W i) 1 W i) r 1 q r ) 1 vi vi ) n vi vi q 0 1 R i S q) vi q r 0 R i S vi q) q r 0 n R i S q) vi q r 0 R i S vi q) q r 1 0 R i E 0 R ) i n 0 R i E 0 R ) i q Finally the gravitational terms can be written as shown in 7) because Gq) is not a function of the reference velocity q r he SL regressor block related to the i-th link becomes where r r 55) Ẋi) W i) + Z i) 0 R n 1 56) Ẋi) W i) + Z i) 1 R n 3 57) r Ẋi) r W i) r R n 6 58) As a result the Slotine-Li manipulator regressor can again be written juxtaposing the regressor blocks Y r i) as Y r q q q r q r ) Y r 1) Y r n) 59) 5 EXAMPLE 51 Explicit calculation of the Slotine-Li regressor for the planar elbow manipulator A simple two-dof planar elbow manipulator as shown in Fig is considered to test the proposed procedure for the calculation of the SL regressor he manipulator is modeled as two rigid links of lengths a 1 a and masses m 1 m Let p G1 p G1x p G1y p G1z and p G p Gx p Gy p Gz be the position vector of the centroids G i with respect to the origin of DH frame O i for i 1 Moreover let I 1 and I be the inertia tensors relative to the centroids of the two links respectively Finally let q q 1 q be the joint coordinates and let q r q r1 q r be the joint reference velocities

g p y G x G a τ l a y 1 τ q G τ 1 x 1 y l 1 p x q 1 G 1 Figure : Planar elbow manipulator In the chosen coordinate frames the computation of the D-H acobians yields v1 a 1 s 1 0 a 1 c 1 0 v a 1 s 1 a s 1 a s 1 a 1 c 1 + a c 1 a c 1 0 0 0 0 ω1 0 0 0 0 ω 0 0 0 0 1 0 1 1 where c i cos q i ) s i sin q i ) for i 1 and c 1 cos q 1 + q ) s 1 sin q 1 + q ) Considering the definitions of Ẋi) r W r i) and Z i) the terms contributing to the SL regressor block Y 1) for the first link are Ẋ 1) a 1 q r1 Ẋ 1) a1 q 0 r1 0 0 Ẋ 1) 0 0 0 0 0 qr1 0 0 0 r 0 0 0 0 0 0 W 1) 0 1 W 1) 0 3 W 1) r 0 6 Z 1) a1 c 0 1 g y Z 1) c1 g 0 1 y g y s 1 0 0 0 0 where the gravitational acceleration vector with respect to the global frame is g 0 g y 0 and

0 n m is a n m zero matrix Analogously for the second link we get Ẋ ) b6 b 7 b 6 q r q + a b 4 + a 1 q r1 + b 4 b 8 1 b 6 b 7 + b 8 q r1 + a b 4 Ẋ ) a b 4 + b 9 b 3 b 10 b b 9 b b 10 b 3 0 b 9 q r1 1 b 10 b 1 + a b 4 1 b 9 b 1 b 10 q r1 0 where Ẋ ) r W ) Z ) 0 0 0 0 0 0 b4 0 0 0 0 0 b 4 0 1 a W ) 1 a s b 5 a1 c 1 + a c 1 ) g y Z ) a c 1 g 1 y 0 0 0 1 a 1 s b 5 1 a 1 c b 5 0 c1 g y g y s 1 0 c 1 g y g y s 1 0 b 1 q r q 1 + q r1 q b q r q 1 + q r1 + q r ) q b 3 q r1 + q r b 4 q r1 + q r b 5 b 3 q 1 + q r1 q b 6 a 1 a s b 7 q r q 1 + q r1 q b 8 a 1 a c b 9 a 1 c b 10 a 1 s W ) r 0 6 hus referring to 55) 56) 57) and 58) we can write the SL regressor blocks Y r 1) and Y r ) associated to link 1 and he manipulator regressor Y rrr is obtained by juxtaposing these blocks as in 59) Finally considering 19) 0) 1) and 35) we can build the inertial parameters vector as π 1) 0 m 1 π 1) 1 m 1 p G1x m 1 p G1y m 1 p G1z π 1) 1xx 1xy 1xz 1yy 1yz 1zz π ) 0 m π ) 1 m p Gx m p Gy m p Gz π ) xx xy xz yy yz zz π 1) π 1) 0 π 1) 1 π 1) π ) π ) 0 π ) 1 π ) π π 1) π ) 5 Derivation of the classical regressor as a particular case Further we can derive the classical regressor 34) for the planar elbow RR) manipulator Y RR q q q) simply by substituting q r q and q r q in the obtained SL regressor Y rrr q q q r q r ) he classical regressor Y RR q q q) can be also computed by following the algorithm proposed in Section 3 with identical results Adopting the proposed algorithms for the direct formulation of both the classical and Slotine-Li regressor we get a vector π as function of ten inertial parameters per link However some of these parameters have no effect on the dynamics of the manipulator We can determine the parameters not affecting the dynamic model simply inspecting the expressions of regressor blocks Y 1) and Y ) and neglecting the parameters that correspond to the zero columns of

the complete regressor Y RR hus for the considered RR manipulator we have Y 1) y11 y 1 y 13 0 0 0 0 0 0 y 110 0 0 0 0 0 0 0 0 0 0 Y ) y111 y 11 y 113 0 0 0 0 0 0 y 10 y 11 y 1 y 13 0 0 0 0 0 0 y 0 and the parameters that affect the dynamics are m 1 m 1 p G1x m 1 p G1y 1zz m m p Gx m p Gy zz Hence the reduced regressor becomes a 8 matrix without zero columns and it is analogous to the one presented in 1 Par 43) 53 Implementation of the procedure in a software package he results described in this section have been obtained by employing the Wolfram Mathematica M package ScrewCalculus developed by M Gabiccini 13 In this package the main function Regressor returns the expression of the complete manipulator regressor A sample code that produces the classical and Slotine-Li regressors for the planar elbow manipulator with this Mathematica M function is shown below <<ScrewCalculus ScrewCalculus tabq1_ q_ {{a1 0 0 q1 "R" {a 0 0 q "R"; qt_ {q1t qt; qdt_ Dqt t; qddt_ Dqdt t; vt_ {v1t vt; vdt_ Dvt t; g0 {0 -g 0; Y Regressortab@@qtqtqdtqdtqddttg0; YrRegressortab@@qtqtqdtvtvdttg0; he function Regressor returns the classical regressor Y and the SL regressor Yr evaluating the DH table of the manipulator tab the vector of joint coordinates qt its first and second derivative qdt and qddt with respect to time t the vector of reference velocity vt its time derivative vdt time t and the components of the gravitational acceleration vector with respect to the global frame g0 For further details and examples regarding the package please refer to 13 6 CONCLUSIONS A direct formulation of the regressor for a general n-dof manipulator has been derived using the Lagrangian approach in a fresh way In addition a modified algorithm for the special regressor required in the Slotine-Li adaptive control algorithm has been presented he developed algorithms has been implemented in a Mathematica M package which has been successfully tested on different manipulators he availability of closed-form expressions for arbitrary serial manipulators could be employed as a good starting point for further investigations on significant topics such as parameters identifiability and automatic definition of minimal sets of parameters References 1 C G Atkeson C H An and M Hollerbach Estimation of inertial parameters of manipulator loads and links he International ournal of Robotics Research vol 5 no 3 pp 101 119 1986

P Khosla and Kanade Parameter identification of robot dynamics in Proceedings of IEEE Conference on Decision and Control Fort Lauderdale Florida 1985 3 H Kawasaki and K Nishimura erminal-link parameter estimation and trajectory control of robot manipulators in Proceedings of IFAC Robot Control 1985 pp 49 54 4 Craig P Hsu and S Sastry Adaptive control of mechanical manipulators he International ournal of Robotics Research vol 6 no pp 16 8 1987 5 - E Slotine and W Li On the adaptive control of robot manipulators he International ournal of Robotics Research vol 6 no 3 pp 49 59 1987 6 Wolfram Research Inc Mathematica 60 Champaign Illinois: Wolfram Research Inc 007 7 M Gautier and W Khalil Direct calculation of minimum set of inertial parameters for serial robots IEEE ransactions on Robotics and Automation vol 6 no 3 pp 368 373 1990 8 H Mayeda K Yoshida and K Ousuka Base parameters of manipulator dynamics in Proceedings of IEEE Conference on Robotics and Automation Philadelphia 1988 pp 1367 1373 9 W S Lu and Q H Meng Regressor formulation of robot dynamics: Computation and applications IEEE ransactions on Robotics and Automation vol 9 no 3 pp 33 333 1993 10 Yuan and B Yuan Recursive computation of the Slotine-Li regressor in Proceedings of the American Control Conference Seattle Washington 1995 pp 37 331 11 M W Spong and M Vidyasagar Robot Dynamics and Control ohn Wiley & Sons 1989 1 L Sciavicco and B Siciliano Modelling and Control of Robot Manipulators London: Springer 000 13 M Gabiccini ScrewCalculus: a Mathematica Package for Robotics DIMNP University of Pisa http://wwwingunipiit/ d11181/rar/screwcalculusrar 009