Robotica () volume 9, pp. 675 69. Printed in the United Kingdom Cambridge Universit Press Mobile manipulator modeling ith Kane's approach Herbert G. Tanner and Kostas J. Kriakopoulos Control Sstems Laborator, Dept. of Mechanical Engineering, National Technical Universit of Athens, 9 Heroon Poltechniou St., 5 78 Zografou, Athens (GREECE) E-mail: {htanner,kkria}@central.ntua.gr (Received in Final Form: Februar 8, ) SUMMARY A heeled mobile manipulator sstem is modeled using Kane s dnamic equations. Kane s equations are constructed ith minimum effort, are control oriented and provide both phsical insight and fast simulations. The poerful tools of Kane s approach for incorporating nonholonomic motion constraints and bringing noncontributing forces into evidence are eploited. Both nonholonomic constraints associated ith slipping and skidding as ell as conditions for avoiding tipping over are included. The resulting equations, along ith the set of constraint equations provide a safe and complete frameork for developing control strategies for mobile manipulator sstems. KEYWORDS: Manipulator modeling; Kane s approach; Nonholonomic restraints; Control strategies.. INTRODUCTION A mobile manipulator is a sstem composed of a manipulator attached to a mobile base. In this a, the manipulator orkspace is drasticall increased. Due to the motion constraints imposed on the mobile platform, such a sstem is usuall nonholonomic. Nonholonomic constraints reduce the dimension of the state space ithout affecting the sstem s configuration space. Man of the models proposed for mobile manipulator sstems,3 do not include nonholonomic constraints. In reference [] an underater vehicle ith multiple robotic manipulators is modeled. Dnamic modeling is achieved b the use of Kane s dnamic equations. 4 Angular momentum preservation las impose nonholonomic constraints on the URV hich are not included in the model. Khatib et al. 3 model cooperating mobile manipulators, the mobile platforms of hich move on a planar surface. The obtain the dnamic equations using the Lagrange formulation in operational space, 3 assuming that the mobile platforms can move in a holonomic a. Other models include nonholonomic constraints using Lagrange 5 and Neton-Euler formulations. 6 In these models not all kinds of nonholonomic constraints imposed are considered. Chen and Zalzala 6 include the nonholonomic constraint associated ith the no skidding condition. In reference [5] the nonholonomic constraint resulting from the no slipping condition is also included. Constraints are merged ith the sstem equations using Lagrange multipliers. Recentl, Thanjavur and Rajagopala 7 modeled an AGV using Kane equations. The pointed out the merits of using Kane s approach to model vehicles and utilized some of the tools to incorporate nonholonom. The focused on the dnamics of the vehicle individual components (drive and castor heels, drive heel assemblies, etc.) and included the no slipping and no skidding constraints. Modeling is usuall treated as a preprocessing stage for the application of a control strateg. Hoever, careful modeling can remove unnecessar detail and reveal some hidden characteristics of the sstem at hand. Thus, instead of being a mere sstem description, an appropriatel constructed dnamic model can greatl facilitate and guide the control design. In this paper a dnamic model is developed, folloing Kane s approach. Compared to Lagrange formulation, Kane s methodolog involves less arithmetic operations and is thus simpler and faster in simulation. 4 What is more, Kane s equations can be easil brought into closed form hich is best suited for control purposes. The modeling procedure presented in this paper enables one to consider the mobile manipulator as a single sstem. With Kane s methodolog it becomes clear that the dnamic interaction beteen the vehicle and the manipulator does not influence performance and thus need not be compensated. On the other hand the same interaction is brought into evidence since it is useful for ensuring contact stabilit ith the ground. Additional dnamic constraints are provided, involving the ground reaction forces, hich eventuall ield a sufficient set of dnamic stabilit conditions. Dnamic effects on tippover stabilit are considered in reference [8]. This paper etends and enriches the earl results of reference [9] b focusing on the contact conditions, eploiting the nonholonom features of Kane s approach and emphasizing on the role of the interaction forces and torques beteen the vehicle and the manipulator in maintaining contact stabilit ith the ground and finall investigating computational compleit issues. The rest of the paper is organized as follos: In Section e introduce the basic terminolog used in the paper and state the assumptions made. In Section 3 e develop the dnamics of the mechanism and calculate the contributions of each of its components to the dnamic equations. Section 4 presents in dnamic form the constraints imposed on the sstem and in Section 5 issues of phsical insight, computational compleit and comparison to alternative methodologies are discussed. A numerical eample is presented in Section 6 and several simulation cases are analzed. Finall, Section 7 concludes ith the main points made in this paper.
676 Kane s approach. PRELIMINARIES Consider a fied inertial frame, {I } as in Figure. Its and aes la on the horizontal plane. At the mobile platform mass center, a frame {v} is attached. Ais z v is parallel to z I. At this mass center, the central principal inertial frame, {v cpi }, is assigned, hich generall differs from {v}. The point here the manipulator is attached on the platform is labeled and coincides ith the platform point (the distinction beteen the to points ill be justified in the sequel). On each manipulator link the central principal inertial frame is identified. Finall, a frame is assigned at a fied point of the manipulator end effector. The superscript on the right side of an quantit ill denote the rigid bod hich it refers to. The superscript on the left ill denote the reference frame ith respect to hich the quantit is epressed. When omitted, frame {I } is supposed. At the center of each heel j,j=,,of the vehicle, frame { j } is attached (Figure ). Its z j ais remains parallel to the vehicle z v ais, but the frame can rotate around this ais. Each frame is related to the fied frame through a rotation matri R. This rotation matri ill be ritten as, i.e. I j R, to indicate transformation of free vectors epressed in frame { j } to the inertial frame. A rotation matri ill be epressed as j i R= i j i j i z j i j i j i z j z i j z i j z i z j, i being the direction cosine beteen j i and j and ( n i ) n j q j = n i n j q j The coefficient of static friction beteen a heel and the ground is assumed the same for all heels and equal to. When eploiting slipping phenomena, one has to consider the friction conditions beteen a heel and the ground more sstematicall. Tire friction ith relation to traction is an issue of ongoing research and lies beond the scope of this paper. The manipulator is supposed to have n 4=m rotational joints. Define the generalized coordinates as q= [ v v q q m ] T here, are the vehicle planar coordinates in the fied coordinate sstem {I }, is its orientation, is the steering angle of the vehicle and the rest correspond to the manipulator joints. The number of generalized coordinates is n since the constraints imposed are completel nonholonomic. Define the folloing generalized speeds: u 3 = u 4 = u r = q r 4 r=5,,m+4 Fig.. Frame assignment on Mobile Manipulator.
Kane s approach 677 In order to save riting and produce to efficient computer code, e introduce auiliar functions (,,, etc.) representing epressions appearing repeatedl. These auiliar functions are defined in the Appendi. In the sequel it is assumed that the motion of the vehicle is restricted to the horizontal plane. It is also assumed that each link on the manipulator rotates relativel to the previous link in the chain, onl in a direction parallel to one of its principal aes of inertia, named z. 3. RIGID BODY CONTRIBUTIONS In this section the kinematics and dnamics of each rigid bod in the mechanism ill be briefl discussed in order to determine the contribution of each bod to the dnamics of the sstem. The detailed derivations have been omitted to preserve the more general perspective, hoever all dnamic contributions can be analticall calculated using the auiliar terms given in the appendi. 3.. Vehicle heels From Kane s approach it follos that all reaction forces eerted on the heels make no contribution to dnamics. Once the nonholonomic motion constraints have been included, due to the motion being planar the ground can be simpl ignored. Each heel is being modeled as a rotating disk. At the center of each heel j a reference frame { j } is attached. The aes are parallel to the central principal inertial aes of the heel: ais j points toards the direction of the heel linear velocit; j is the rolling ais and z j is vertical. A heel has to degrees of freedom. One allos rotation about j ais (Figure ) and is controlled b an input torque j d. The second degree of freedom corresponds to the steering angle of each heel j. Normall, all steering angles are coupled to achieve kinematic compatibilit. The value of onl one of them and the equations of the steering mechanism are sufficient to describe them all. Therefore, e can assume that j = j( ), here is one of them, named the steering angle. The velocit in {I } of frame s { j } origin can be epressed as v j =ẋ v I +ẏ v I + (z v j ), here j is the position vector from the origin of {v} to the contact point of the heel ith the ground (Figure ). The no skidding condition states that there should be no velocit component normal to the heel plane. Epressing v j.r.t. the heel frame this statement is equivalent to =ẋv sin( j + ) ẏv cos( j + ) v j sin j + v j cos j for an j {,,} () Under the no skidding condition the heel velocit reduces to v j =[ 3j u + 4j u ] j, () hile the no slipping constraint can be epressed as: v j = j r j z j (3) here r j is the heel radius. Among the j heels, identif the one hich determines the steering direction and denote that angle b o, so as to clearl distinguish it from the rest. The orientation angle of the vehicle ill then be epressed in terms of that particular angle, =ẋv sin( o + ) ẏ v cos( o + ) according to ():. No let u v o sin o + v o cos o = ẋ v and u v o sin o + v o cos o = ẏ v. v o sin o + v o cos o Then, u 3 = =u sin( o + ) u cos( o + ) (4) The angular velocit of the heel is ritten: j = [ r j 3j u + 4j u ] j +[ 3 + j u 4 ]z j (5) On each heel the folloing contact and field forces (and torques) are eerted: the reaction from the ground, F j, applied at the point of contact beteen the heel and the ground, the reaction from the vehicle bod F vj, the control input torques T j, both applied at frame s {j } origin and the inertial forces and torques. The ground reactions and the vehicle-heel interaction forces, do not contribute to the dnamic equations. The onl contributing generalized forces are the input torques and the inertial forces and torques. B setting j = 3j u + 4j u + ( +)j and j z = u + u + j u 4 + ( +3)j, the generalized inertia and active forces hich contribute to the dnamic equations can be epressed:
678 Kane s approach T* j= j 3j m j j m j (r j) z 4, T* j= j 4j m j j m j (r j) z, 4 T* 4 j= j m j (r j) z j, F* 4 j= m j j 3j, F* j= m j j 4j The contributions of the input and inertial forces and torques to the generalized active forces are (T j s ) = j s, (T j s ) = j s, (T j s ) 4 = j j s, (T j d ) = 3 j r j d, j (T j d ) = 4 j r j d, j here j s and j d are the steering and driving torques eerted b the motors on the heels, respectivel. 3.. Vehicle bod The vehicle bod moves horizontall in a nonholonomic fashion, subject to the combined effect of the forces eerted b the heels and the forces eerted b the attached manipulator. The heel forces are included in the form of reactions to the steering and traction torques eerted on the heels. At the mass center of the vehicle bod a reference frame {v} is attached. Ais v is aligned ith the direction of motion hen the steering angle is zero. Ais z v is vertical and v completes the coordinate sstem. On the vehicle bod the folloing contact and field forces and torques arc eerted: the reaction forces b the heels, a force and torque b the manipulator, the vehicle eight, the inertial forces and torques and the reactions to the driving and steering torques. Among them, the reaction forces b the heels, the force and torque applied b the manipulator and the eight of the vehicle do not contribute to the generalized active forces. Let I v be the vehicle s inertia moment tensor and define: T* v = ( u + u + + ) zv I v ( vcpi 3 ) zv zv vcpi z (I v I v vcpi z ) T* v = ( u + u + + ) zv I v ( vcpi 3 ) zv zv zvcpi (I v vcpi z I v ) T* v z = ( u + u + + ) zv I v z vcpi z ( 3 ) zv zv vcpi (I v I v vcpi ) Let the reaction torques b the heels and the interaction forces eerted b the manipulator be denoted, respectivel, T j s = j s z j T j d = j d j F =F v +F v +F z z v T =T v +T v +T z z v Then contribution of the mobile platform to the generalized inertial forces can be ritten and the contribution to the generalized active forces T* v = T* v vcpi v z T* v vcpi v z T* v z zvcpi z v T* v = T* v vcpi z T* v v vcpi z T* v v z zvcpi F* v = m v ( o u + + ) o F* v = m v ( o u + + ) o z v (T j s ) = j s F =F +3 +F +5 (T j s ) = j s T =T z T =T z F =F +4 +F +6 3.3. MANIPULATOR BASE The manipulator base is rigidl attached to the mobile platform. It does not possess an independent degree of freedom and could be considered a part of the mobile platform. The reason for eamining it separatel, despite computational cost, is the fact that for stationar manipulators the inertial parameters of their base are tpicall ignored and are usuall unavailable. On the other hand, mobile platforms are manufactured separatel. As a result their inertial characteristics include onl the bod and the heels. The unavailabilit of information that ould enable the integration of the manipulator base to other structural components necessitates its independent stud. The interaction forces beteen the manipulator base and the vehicle on hich it is attached to do not contribute to the dnamics and therefore do not appear in the equations. Hoever, these forces are useful in determining and maintaining contact stabilit of the heels. In order to bring into evidence, the interaction forces beteen the manipulator and the
Kane s approach 679 platform e assume that there is a virtual relative movement beteen the to bodies, epressed in terms of some additional fictitious generalized speeds. At the mass center of the manipulator base, c, a reference frame {} is attached. This frame is aligned ith frame {v}. On the manipulator base the folloing forces and torques are eerted: inertial forces and moments, reactions forces b the mobile platform, the reaction torque b the first link of the manipulator, and the base eight. The base eight contributes because of the virtual motion of base ith respect to the platform. Let I be the base s inertia moment tensor and define: F* = +3 u + +4 u + +3, F* = +5 u + +6 u + +4 T* = ( u + u + + ) zv I cpi ( 3 ) zv T* = ( u + u + + ) zv I cpi ( 3 ) zv z T* z = ( u + u + + ) zv z I cpi z ( 3 ) zv cpi zv cpi zv cpi zv z cpi (I I z) cpi (I z U ) cpi (I U ) The interaction forces and torques eerted b the platform to the base are epressed as F = F v F v F z z v T = T v T v T z z v hereas the reaction torque b link and the eight is T = z and G =m gz I, respectivel. The contribution of the inertial moment to the generalized inertial forces ill be T* = T* zv T* m+5= T* T* m+7= T* cpi v cpi z v cpi T* T* T* zv cpi v cpi v cpi T* z zv zcpi T* = T* zv T* z zcpi v T* m+6= T* T* z zcpi z v cpi v cpi T* T* zv cpi v The contribution of the inertial force to the generalized inertial forces ill accordingl be F* = m [F* +3 +F* +5 ] F* = m [F* +4 +F* +6 ] F* m+8= m F* F* m+9= m F* The base eight, the interaction forces and the reaction to the torque on link contribute b: cpi T* z T* z zcpi v zv z cpi G m+=m g F m+8= F F m+9= F F m+= F z T = T z T = T z T m+5= T T m+6= T T m+7= T z T = zv z T = zv z T m+5= v z T m+6= v z T m+7= v z F = F +3 F +5 F = F +4 F +6 3.4. First manipulator link Within Kane s methodolog it is pointed out that out of all forces and torques acting on a link, onl the input torque, the eight and the inertial forces and moments contribute. The reaction forces and torques beteen neighboring links do not contribute to dnamics and can be ignored. This observation results in significant savings in terms of computational and derivation compleit. Frame {} is attached to the mass center of the first manipulator link. Aligned to the principal inertial frarne, it rotates relative to the manipulator base around ais z. On the first link of the manipulator the folloing are applied: inertial forces and moments, the link eight, the input torque and the reaction to the input torque applied to the net link. Define the terms: F* = +9 u + + u + + u 5 + +5 T* = ( 4 u + 5 u + +4 )I (I I z) F* = + u + +3 u + +4 u 5 + +6 T* = ( 6 u + 7 u + +5 )I (I z I ) F* z = +5 u + +6 u + +7 T* z = ( 8 u + 9 u + u 5 + +6 )I z (I I ) The input torque of link and the link s eight are, respectivel, T = z and G =m gz i.
68 Kane s approach The contribution to the generalized inertial forces for link ould be F* = m [F* +9 +F* + +F* z +5 ] T* = T* 4 T* 6 T* z 8 F* = m [F* + +F* +3 +F* z +6 ] T* = T* 5 T* 7 T* z 9 F* 5 = m [F** + +F* +4 ] T* 5 = T* z F* m+5= m [F** + +F* +7 +F* z +3 ] T* m+5= T* + T* +4 T* z F* m+6= m [F** + +F* +8 +F* z +4 ] T* m+6= T* + T* +5 T* z F* m+7= m [F** +3 +F* +9 +F* z +5 ] T* m+7= T* +3 T* +6 T* z +7 +8 +9 F* m+8= m [F** +4 +F* + +F* z +6 ] T* m+9= m [F* +5 +F* + +F* z +7 ] F* m+= m [F** +6 +F* + +F* z +8 ] On the other hand, the contribution of the eight is calculated as follos: G =m g( z +9+ I z ++ I z z +5) I G 5=m g( z ++ I z +4) I G m+6=m g( z ++ I z +8+ I z z +4) I G m+8=m g( z +4+ I z ++ I z z +6) I G m+=m g( z +6+ I z ++ I z z +8) I G =m g( z I ++ z I +3+ z z I +6) G m+5=m g( z I ++ z I +7+ z z I +3) G m+7=m g( z I +3+ z I +9+ z z I +5) G m+9=m g( z I +5+ z I ++ z z I +7) hile the input torque contributes b T = 8 T = 9 T 5= T m+5= v z T m+6= v z T m+7= v z 3.5. Links to m Links to m are analzed as link and recursive epressions for the kinematic quantities are derived. In this a, there is no constraint to the number of the manipulator joints that can be included in the model. At the mass center of link i, for i=,,m frame {i} is attached. It is aligned to the principal inertial frame. Link i rotates.r.t. link i along ais z i. For the rest of the links the treatment is similar. We first identif the forces and torques acting on each link: the inertial forces and moments, the link s eight and the input torque. Define: F* i = i u + i + u + i + u 5 + + i +i u i+3 + i +i+ u i+4 + i + F* i = i +i+ u + i +i+3 u + i +i+4 u 5 + + i +i+ u i+3 + i +i+3 u i+4 + i + F* i z = i +i+4 u + i +i+5 u + i +i+6 u 5 + + i +3i+4 u i+3 + i +3 T* i =( i u + i + u + i + u 5 + + i + u i+3 + i +)I i i +3i+4 i +3i+5(I i I i z ) T* i =( i +i+ u + i +i+ u + i +i+3 u 5 + + i +i+ u i+3 + i +)I i i +3i+5 i +3i+3(I i z I i ) T* i z =( i +i+ u + i +i+3 u + i +i+4 u 5 + + i +3i+ u i+3 + u i+4 + i +3)I i z i +3i+3 i +3i+4(I i I i ) The input torque and the eight of the link are T i = i z i and G i =m i gz I, respectivel. The contributions of the inertia moments and forces to the generalized inertial forces are: [T* i T* i T* i 5 T* i i+3 T* i i+4 T* i m+5 T* i m+6 T* i m+7] T = i i +i+ i +i+ i + i +i+ i +i+3 i + i +i+3 i +i+4 i +i i +i+ i +3i+ i + i +4 i +7 i + i +5 i +8 i +3 i +6 i +9 T T* i T* i T* i z,
Kane s approach 68 [F* i F* i F* i 5 F* i i+3 F* i i+4 F* i m+5 F* i m+6 F* i m+7] T = m i i i +i+ i +i+4 i + i +i+3 i +i+5 i + i +i+4 i +i+6 i +i i +i+ i +3i+4 i +i+ i +i+3 i + i +7 i +3 i + i +8 i +4 i +6 i + i +8 T F* i F* i F* i z, The link eight and the input torque contribute to the generalized active forces b: [G* i G* i G* i 5 G* i i+3 G* i i+4 G* i m+5 G* i m+6 G* i m+7] T =m i g i i +i+ i +i+4 i + i +i+3 i +i+5 i + i +i+4 i +i+6 i i +i+ i +3i+4 i + i +i+3 i + i +7 i +3 i + i +8 i +4 i +6 i + i +8 T i z I i z I zi z I, T i i+4= i 3.6. End effector The end effector is analzed onl as being the point here the load is applied. It makes no inertial contribution b itself; that contribution has been included through the analsis of the preceding links. Thc end effector, ee, is considered a part of the last link in the manipulator chain. Therefore, its angular velocit ill be m, since the point belongs to the rigid bod m. If an eternal load is applied at the end effector of the manipulator, then this can be represented b a couple torque and a force T ee =T ee m +T ee m +T ee z z m F ee =F ee m +F ee m +F ee z z m The contribution of this torque and force to the generalized active forces ould be [T ee T ee T ee 5 T ee m+7] T = A T m M T m T ee T ee T ee z, [F ee F ee F ee 5 F ee m+] T = B ee [ ] ee N T m F ee F ee F ee z, 3.7. Dnamic equations The sstem dnamic equations can no be derived b adding the contributions of each rigid bod (F*+F+T*+T)= There ill be as man equations as generalized speeds. For generalized speeds u, u and u 4 : = j= = j= = j= [T* j +F* j +(T j d ) ]+T* v +F* v +T* +F* +F* +T* [T* j +F* j +(T j d ) ]+T* v +F* v +F* +T* +F* +T* m j j s j j (r j ) z 4 j m +G+ i= m +G+ i= The dnamic equations that correspond to the manipulator generalized speeds are [T* i +F* i +G i ]+T ee +F ee (6a) [T* i +F* i +G i ]+T ee +F ee (6b) (6c)
68 Kane s approach m = r 4 + (T* r i +F* r i +G i r )+T ee r +F ee r, r {5,,m+4} (6d) i=r 4 The dnamic equations that involve the non-contributing forces that have to come into evidence are: =T* m+p+t m+p+g m+p+t* m+p+f* m m+p+ i= (T* i m+p+f* i m+p+g i m+p)+t ee m+p+f ee m+p, p=5,6,7 (6e) =F* m+s+f m+s+g m+s+f* =m g+f m++g m++f* m m+s+ i= m m++ i= (F* i m+s+g i m+s)+f ee m+s s=8, 9 (6f ) (F* i m++g i m+)+f ee m+ (6g) In developing the above dnamic equations, care has been taken so that generalized accelerations appear eplicitl. This facilitates the transformation of the equations to the closed form: M u+f=b (7) here M is the inertial matri, u is the vector of the generalized accelerations, F is the vector of the Coriolis, centrifugal and gravit terms and B is the matri that multiplies the input variable vector. In reference [] eplicit formulas for such a transformation are given, although this is actuall a matter of algebraic manipulation. 4. CONSTRAINTS In Section 3, to kinematic constraints have been stated. The rolling ithout slipping constraint, v j = j r j j =ẋv j cos( j + )+ẏ v sin( j + )+ ( v j and the no skidding constraint for the heels: r j sin j v j cos j ) v j =ẋv sin( j + ) ẏv cos( j + ) = v j sin j + v j cos j for an j {,,} Finall, the relation beteen the velocities of a heel center and the vehicle mass center is v v =v j z v j The above equation can be used to eliminate another generalized speed, although this option has not been utilized in this paper. The relation can be equivalentl described b an equation in the form q=sz, here q is the vector of the remaining generalized speeds, ż is the vector of the independent generalized speeds and S is a matri of an appropriate dimension. B differentiating and left-multipling (7) b S T, e obtain: S T MSż+S T (MṠz+F)=S T B (8) Equations (8) are the final, reduced form of the dnamic model of the sstem, ith all constraints included. Nonholonomic constraints have been integrated ithin the model equations, ithout the need for Lagrange multipliers. This can ensure that in theor the vehicle trajector is nonholonomic and that the platform heels roll ithout slipping. What ensures, hoever, is that in real applications the constraints ill be respected? In fact, velocit conditions such as the one used above are onl necessar and not sufficient, because the describe the result and not the cause. In this section, e treat this problem b completing the set of nonholonomic constraints b additional epressions involving constraint reaction forces at the heels. We specif the domain of admissible values for the constraint forces and indicate ho friction forces ill not saturate and ho tipping over can be avoided.
Kane s approach 683 The set of constraint equations follos. The no slipping condition is adopted from reference [] hile the tip-over criterion used is the Force-Angle stabilit measure. 8 Rz = I j= F j z I (vertical balance) F j z =F j z j r j F j j= j d (driving torques) + F j z j F j (no slipping) R+ F j F j ( v v v )= (no skidding) =min( i ) R > (no tipover) j= (9) here i is the angle measure associated ith each tip-over ais, i, given b i =sign {(ˆl i ˆf* i ) â i } arccos(ˆf* i ˆl i ), the reaction forces eerted on the heels are denoted b F j, R is the resultant of all contact and field forces eerted on the platform given b R=m v (g a v )+ j= m j (g a j )+F, is the coefficient of static friction beteen a heel and the ground, and the rest of the smbols are defined as f* i =f i +ˆl i n i l i ˆl i = l i l i ˆf* i = f* i f* i â i = a i a i l i =(I â i â T i )( v i+ r i zv v p vcm ) f i =(I â i â T i ) ˆR n i = j= a i = i+ i T* j T*v T* +T Equations (6) along ith constraints (9) constitute a dnamic model of a mobile manipulator under load ith the full set of constraints imposed on the sstem. 5. DISCUSSION 5.. Phsical insight Phsical insight stems from the fact that in the final epressions, one is capable of distinguishing the forces and torques that influence the dnamic behavior of the mechanism. Take, for eample, (6), in the case of a mobile manipulator ith to horizontal links and no load (see Section 6): = j= = j= = j= [T* j +F* j +(T j d ) ]+T* v +F* v +T* +F* +F* +T* +T* +F* [T* j +F* j +(T j d ) ]+T* v +F* v +T* +F* +F* +T* +T* +F* m j j s j j (r j ) z 4 j = +T* 5 +F* 5 +T* 5 +F* 5 = +T* 6 +F* 6 +T* 6 +F* 6 It is evident that the onl contributing forces are the inertial forces/torques and the input torques. The eight does not contribute because the motion is planar. Nor do an nonholonomic constraint forces and therefore the do not have to be calculated. On this issue, the approach departs from conventional methodologies hich require calculation or elimination of the constraint forces. B eamining the sstem from Kane s methodolog viepoint, it is evident that hen having the complete model of the mobile manipulator the interaction forces beteen the manipulator and the mobile platform do not contribute to dnamics
684 Kane s approach and therefore the have no impact on the sstem s performance. Therefore, contrar to eisting approaches, the control design need not focus on cancellation of this interaction. Such control approaches 5 could be justified ithin a frameork of decentralized control, but cannot promise improved dnamic performance. What is also clear is the effect of ignoring the friction at the heels: an infinitesimal steering torque ill cause an undamped rotation of the heels. The most important characteristic of the nonholonomic attribute of the sstem, reflected on the dnamic equations is ith no doubt the lack of correspondence beteen the number of degrees of freedom and the number of dnamic equations. 5.. Computational compleit Computational compleit is another important issue. Kane and Levinson 4 modeled a Stanford Manipulator and reported feer multiplications and additions than an other approach. Kane, hoever has not described analticall the compleit of his computational algorithm. We have derived an upper bound for the calculations our model requires: multiplications: 37.5m +44.5m+88+8, additions: 3.5m +39.5m+6+, here m is the number of manipulator links and is the number of the vehicle heels. Both the number of multiplications and additions belong to O(m ). 5.3. Comparison to alternative methodologies The algorithm is obviousl superior to Lagrange formulations as far as computational compleit is concerned, since the Lagrange computational algorithm is in O(m 4 ). As far as Neton-Euler formulation is concerned, it must be noted that the parabolic curves presented above lie belo the Neton s line for reasonable numbers of sstem degrees of freedom. To make a simple comparison, consider the case of a vehicle ith four heels one manipulator link. The number of rigid bodies in the sstem is 6. Our algorithm ill require in the orst case, Kane: 679 multiplications 494 additions To estimate the number of calculations needed for a Neton-Euler algorithm, e borro the polnomials derived for the case of a Puma manipulator ith 6 links 3 hich gives: Neton: 678 multiplications 597 additions Moreover, Neton-Euler formulation ehibits difficulties in dealing ith reaction forces at the heels. Its recursive nature requires the calculations of all forces and torques eerted on the sstem. Hoever, the distribution of the reaction forces to the heels is unknon. 4 Finall, it must be emphasized that our equations are not optimized in terms of compleit, and are presented in the most general form. Care has been taken so that generalized accelerations appear eplicitl and the equations can easil be brought into the form Mẍ+C()ẋ+K()=F, hich is suitable for control. The same cannot be said, hoever, for Neton-Euler equations. Fig.. Mobile manipulator used in the eample.
Kane s approach 685 6. NUMERICAL VERIFICATION This eample is a three heel, to link mobile manipulator (Figure ). The geometric and dnamic parameters used are given in Table I. Simulations ere conducted in MATLAB using an ode5s integrator. Input torques ere set to zero. For a number of motion cases, Figures 3 5 depict the motion of the mobile manipulator, the total energ of the sstem, the friction forces and the influence of forces and torques on stabilit against tip-over. Monitoring the total energ of the sstem can help verif the validit since in the absence of input torques, the Hamiltonian of the sstem should remain invariant. In the planar eamples, this Hamiltonian reduces to the kinetic energ of the sstem. As can be seen in all simulations, the kinetic energ of the mechanism remains constant, impling that the sstem model is consistent. The calculation of friction forces serves in determining stabilit margins against slipping and skidding. Friction forces are directl related to velocities and accelerations, as shon in Section 4. Ensuring small values for the necessar centripetal force can safeguard against the risk of skidding. Modeling the characteristics of tire friction is quite demanding and is still an issue of research in progress. Friction models,5 can serve toards this direction and significant effort is currentl devoted to etending their applicabilit and generalit. Another important measure for mechanical stabilit is the tip-over stabilit measure. Normall this measure should take positive values. Negative values indicate that the sstem is on the verge of tipping over, having lost contact of some of its heels ith the ground. Whether tip-over ill occur depends on the time interval for hich the measure remains negative and the configuration of the sstem in Table I. Geometric and dnamic parameters used in the eample sstem. Geometric Parameters (m) l =.3 l = l =.3 v r = v r = v r = r c =.5 l =.3 l =.3 l =.3 v r = v r z = r c = r c z =.55 r c =.8 r c = r c z = r =.5 r c = r c z = r c =.4 Inertial Parameters (kg or kg m ) m = m = m =5 I v =.667 I v z =3. I =.8 I =.35 I =.759 I z =.67 m =5 m v =4 I v =.667 I =.8 I z =.56 I z =.73 I =.67 I = Fig. 3. Motion in straight line ith aligned links.
686 Kane s approach Fig. 4. Circular motion of the first link. terms of the abilit of gravitational stabilizing forces a moment at that particular configuration to bring the mechanism back in full contact ith the ground. 8 In real applications, this quantit should alas remain positive. In simulations hoever e have the advantage to let this measure var freel in order to determine the kind of motion that is most likel to cause tip-over. In this perspective, in the tpe of motion depicted in Figures 4 and 5 the risk of tip-over is substantial. In Figure 3 the mobile platform has an initial translational velocit and moves in a straight line hile the to links are aligned ith the direction of motion. As epected, due to the straight line motion of all components, no centripetal forces are developed. The tip-over stabilit measure is positive indicating no risk of tip-over. Since no input torques are applied, the kinetic energ remains constant at all times and the motion produced in the simulations verifies that the robot maintains its straight motion. In Figure 4 the base is stationar and the first link rotates hile the second joint is unactuated (passive). This motion generates inertial forces hich enforce nonholonomic motion on the base. The nonholonomic motion is depicted in Figure 4 in the horizontal oscillation of the platform indicated b the shaded region. The dnamic effect of the rotation of the passive link is depicted in the varing velocit of the first link (successive plots correspond to constant time steps). Centripetal forces accelerate the links and produce the irregular spacing beteen each link configuration. This motion can cause tip-over during the time periods indicated in Figure 4 b the thick lines on the time ais of the graph representing the tip-over stabilit measure. During the time hen the first link angle is close to / and 3 / the centrifugal forces are sufficient to cause tip-over. Finall, in Figure 5, the base moves on a circular path hile the manipulator links are unactuated. This case is useful to observe ho the motion of the links affects the mobile platform s speed and ho the centrifugal forces induced b the circular motion of the platform cause link rotation. Thc influence on platform speed is evident from the irregular spacing beteen successive platform locations. Tip-over can occur during the time periods indicated b the thick lines on the time ais of the tip-over stabilit measure graph, corresponding to the configurations here the first link is ovcr the right side of the mobile platform. 7. CONCLUSIONS AND ISSUES FOR FURTHER RESEARCH In this paper a complete model for a mobile manipulator sstem has been constructed, consisted of a set of dnamic equations and a set of constraint equations. The model is formulated folloing Kane s approach hich is advantageous to conventional approaches. Within that frameork, the importance of the dnamic interaction beteen the vehicle and the attached manipulator for stabilit as emphasized and the methods for bringing such an interaction into evidence ere eploited. Monitoring the reaction forces from the ground is necessar in order to secure the
Kane s approach 687 Fig. 5. Circular motion of the platform ith unactuated arm. sstem from tipping over and to guarantee that nonholonomic constraints are respected. It is pointed out that conventional velocit nonholonomic constraint equations are not sufficient to impose a nonholonomic motion unless accompanied ith their dnamic counterparts, hich are formall epressed and presented in this paper. Current issues of interest include modeling multiple cooperating mobile manipulator sstems to enable effective control strategies. References. G. Campion, B. d Andrea Novel, and G. Bastin, Modelling and state feedback control of nonholonomic mechanical sstems, Proceedings of the 99 IEEE Conference on Decision and Control (December, 99).. T. J. Tarn, G. A. Shoults, and S. P. Yang, Dnamic model for an underater vehicle ith multiple robotic manipulators, Pre-Proc. 6th Int. Advanced Robotics Program, Toulon-La Sene (996) pp. 3. 3. O. Khatib. K. Yokoi, D. Ruspini, R. Holmberg, A. Casal, and A. Baader, Force strategies for cooperative tasks in multiple mobile manipulation sstems, Int. J. Robotics Research (7), 333 34 (996). 4. Thomas R. Kane and David A. Levinson, The use of Kane s dnamical equations in robotics, Int. J. Robotics Research (3), 3 (October, 996). 5. Yoshio Yamamoto, Control and Coordination of Locomotion and Manipulation of a Wheeled Mobile Manipulator, PhD thesis (Dept. of Computer and Information Science, School of Engineering and Applied Science, Universit of Pennslvania, Philadelphia, PA, August 994). 6. M. W. Chen and A. M. S. Zalzala, Dnamic modelling and genetic-based motion planning of mobile manipulator sstems ith nonholonomic constraints, Research Report 6 (Robotics Research Group, Dept. of Automatic Control and Sstems Engineering, Universit of Sheffield, UK, September 995). 7. K. Thanjavur and R. Rajogopalan Ease of dnamic modelling of heeled mobile robots (WMRs) using Kane s approach, IEEE Int. Conf. Rob. and Autom., Albuquerque, Ne Meico (997) pp. 96 93. 8. E. G. Papadopoulos and D. A. Re, A ne measure of tipover stabilit margin for mobile manipulators, IEEE Int. Conf. Robotics and Automation, Minneapolis, MN (April, 996). 9. H. G. Tanner and K. J. Kriakopoulos, Modeling of a mobile manipulator ith the use of Kane s dnamical equations, IFAC Meeting on Intelligent Autonomous Vehicles, Madrid (March, 998) pp. 4 4.. C. Canudas de Wit and P. Tsiotras, Dnamic tire friction models for vehicle traction control, Proc. of the 999 IEEE International Conference on Decision and Control, Phoeni, Arizona (December, 999) pp. 3746 375.. R. Thomas Kane and A. David Levinson, Dnamics: Theor and Applications(McGra-Hill, 985).. Yoshihiko Nakamura, Advanced Robotics: Redundanc and Optimization (Addison-Wesle, 99). 3. K. S. Fu, R. C. Gonzalez, and C. S. G. Lee, Robotics. Control, Sensing, Vision, and Intelligence (McGra-Hill, 989). 4. S. Dubosk and E. E. Vance, Planning mobile manipulator motions considering vehicle dnamic stabilit constraints, Proc. IEEE Int. Conf. on Robotics and Automation (989) pp. 7 76. 5. H. Olson, K. J. Aström, C. Canudas de Wit, M. Gäfrert, and P. Lischinsk, Friction models and friction compensation, European Journal of Control 4, 76 95 (December, 998).
688 Kane s approach APPENDIX Let m be the total number of manipulator links and i> =3+9m+ 3(m )m, =3m +7m+, =3m +m+4, =3m +3m+8, =3m +3m+8, i =4+9(i )+ 3(i )(i ), i =3m +m+5+3i, i =5i 6+ 3(i )(i ), i =3m +3m+9+9i, i =+9m+ 3(m )m +(i )+ 3(i )(i ), i =3m +7m++3i, i =3m +3m+8i, i = 3(m )(m ) +8m +9i, c =cos, s =sin, c j =cos j, o = v o s o + v o c o, c o =cos o, s o =sin o, c j =cos( j + ), 3j = o c j j, s j =sin j, 9 = zv z, s j =sin( j + ), j = v j c j v j s j, 5 = zv, =cos( o + ), =sin( o + ), 4j = o s j j, j = j, 3= u + u, 4 = zv, 6 = zv, = 4 u + 5 u 8 = zv z, 7j = v j s j + v j c j, 8o = v o c o v o s o, 7 = zv, = 6 u + 7 u = 8 u + 9 u +u 5 + = o u, +4 = o s v r, +3 = o c v r, +7 = +3 u + +4 u, +8 = +5 u + +6 u, +9 = +5 u + +6 u, +6 = v z 3+ v z 4+ r c 5 r c 7, +8 = + u + +3 u + +4 u 5, +7 = +9 u + + u + + u 5, +9 = v + v + r c z 6 r c 78, + = o u, + = v 3+ v 4+ r c z 7 r c 9, +5 = v r o s, + = v + v + r c z 8 r c 4, +6 = c r + o c, +3 = v 3+ v 4+ r c 9 r c z 5, + = r c, +5 = v z + v z + r c 4 r c 6, +4 = r c, = +3 v r, = +5 + v r, 3 = +4 v r, 4 = +6 + v r, 5 = r c z 6 r c 8 + +9, 6 = r c 8 r c z 4 + +, 7 = r c 4 r c 6 + +5, 8 = r c z 7 r c 9 + +, 9 = r c 9 r c z 5 + +3, = r c 5 r c 7 + +6, = r c + l+, = r c + +4,
Kane s approach 689 + =( u u ) o 3 +Z u 4, ( +)j = 8o [u c j +u s j ] o 7 j j 3 u 4 j + + o ( j u 4 + 3 )(u c j u s j ), ( +3)j = + + j u 4, +4 =( zv ) q 3 u 5 + zv +, +5 =( zv ) q 3 u 5 + zv +, +6 =( zv z ) q 3 u 5 + zv z +, o + = 8o u 4 u 3 o u, o +3 = 8o u 4 v r + + 3 ( o +8 ), o + = 8o u 4 u + 3 o u, o +4 = 8o u 4+ v r + 3 ( o +7 ), +5 = 3 + +9 +8, +6 = 4 + +7 +9, +7 = 5 + +8 +7, =c u +s u, =c u s u, 3 ={[( v ) q +( v ) q ]u +[( v ) q 3 +( v ) q 4 ]u }u 5 + v 8o o u 4 + o 3 + v o 8o u 4 o 3 +[(v r + v r ) v 4 ={[( v ) q +( v ) q ]u +[( v ) q 3 +( v ) q 4 ]u }u 5 + v (v r + v r ) ] ++ r c z +5 r c +6, 8o o u 4 + o 3 + v o 8o u 4 o 3 +[(v r + v r ) v 5 ={[( v z ) q +( v z ) q ]u +[( v z ) q 3 +( v z ) q 4 ]u }u 5 + v z (v r + v r ) ] ++ r c z +6 r c +4, 8o o u 4 + o 3 + v z o 8o u 4 o 3 +[(v r + v r ) v z (v r + v r ) z z ] ++ r c +4 r c +5, + = +4 r c z +7 r c v r z v +v r zv, + = +5 r c z +8 r c + v r z v +v r zv, +3 = +6 r c z +9 r c v r v +v r v, +7 = +7 r c + r c z v r z v +v r zv, +8 = +8 r c + r c z + v r z v +v r zv, +9 = +9 r c +3 r c z v r v +v r zv, +3 = + r c +4 r c v r z v z +v r zv, z +4 = + r c +5 r c + v r z v z +v r zv, z +5 = +3 r c +6 r c v r v z +v r v, z +4= v, +5= v, +6= zv, += v, += v, += zv, +6= v z, +7= v z, +8= zv z, += v, += v, +3= zv, +4= v, +5= v, +6= zv, +7= v z, +8= v z, +9= zv z,
69 Kane s approach A i =( i i R) A i, A i + = A i [u u u 5 u i+4 ] T, [ ] i =L ci i A i +B T i, B i =[ i i RL ici ] [ ]T i A T i T, B + i =B i [u u u 5 u i+4 ] T, i = (i i R) q i A + i +( i i R) i, [ ] i =N i T +L ci i M i [ ] i =L ci i i +[ ] i, [ ] i = (i i R) q i [ ] i [u u i+3 ] T u i+4+( i i R)[ ] i +L ici i, i =[ ] i +A + i B + i, M i =( i i R)M i, N i =( i i R)[ ] i +L ici M i, N i =( i i R)N i,