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- 1. Multibody System Dynamics 2: 317–334, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands. 317 A New Approach for the Dynamic Analysis of Parallel Manipulators JIEGAO WANG and CLÉMENT M. GOSSELIN Département de Génie Mécanique, Université Laval, Québec, Québec G1K 7P4, Canada (Received: 26 November 1997; accepted in revised form: 27 May 1998) Abstract. A new approach for the dynamic analysis of parallel manipulators is presented in this paper. This approach is based on the principle of virtual work. The approach is ﬁrstly illustrated using a simple example, namely, a planar four-bar linkage. Then, the dynamic analysis of a spatial six-degree-of-freedom parallel manipulator with prismatic actuators (Gough–Stewart platform) is performed. Finally, a numerical example is given in order to illustrate the results. The approach proposed here can be applied to any type of planar and spatial parallel mechanism and leads to faster computational algorithms than the classical Newton–Euler approach when applied to these mechanisms. Key words: parallel manipulators, parallel mechanisms, dynamics, virtual work, simulation, control. 1. Introduction Parallel manipulators have received more and more attention over the last two decades (see, for instance, [1, 5, 6, 8, 9, 13, 14]). Among other issues, the dynamic analysis of parallel manipulators has been studied by several authors (see, for in- stance, [2–4, 7, 10, 12, 15, 18]). The most popular approach used in this context is the Newton–Euler formulation, in which the free-body diagrams of the links of the manipulator are considered and the Newton–Euler equations are applied to each isolated body. Using this approach, all constraint forces and moments between the links are obtained. Although the computation of such forces and moments is useful for the purposes of design, they are not required for the control of a manipulator. In [11], the dynamic analysis of a three-degree-of-freedom parallel manipulator using a Lagrangian approach is presented. However, because of the complexity of the kinematic model of the spatial parallel manipulator, some assumptions have to be made to simplify the expressions of the kinetic and potential energy. There- fore, this approach is not general and efﬁcient for the dynamic analysis of parallel mechanisms or manipulators. In this paper, a new approach based on the principle of virtual work is proposed. First, the inertial “force” and “moment” are computed using the linear and angular accelerations of each of the bodies. Then, the whole manipulator is considered to Author for correspondence.
- 2. 318 J. WANG AND C.M. GOSSELIN be in “static” equilibrium and the principle of virtual work is applied to derive the input force or torque [16]. Since constraint forces and moments do not need to be computed, this approach leads to faster computational algorithms, which is an important advantage for the purposes of control of a manipulator. 2. Illustration of the Approach The well known planar four-bar linkage is represented in Figure 1. It consists of three movable links. The links of length l1, l2 and l3 are respectively the input link, the coupler link and the output link and their orientation is described respectively by angles θ, α and φ. If φ, ˙φ and ¨φ are known, the solution of the inverse dynamic problem consists in ﬁnding the torque τ that is required to actuate the input link to produce the speciﬁed trajectory. In this section, the dynamic analysis of this one- degree-of-freedom mechanism using the approach of virtual work is performed in order to illustrate the application of the approach. 2.1. COMPUTATION OF THE INERTIAL FORCES AND MOMENTS OF EACH LINK Following d’Alembert’s principle [17], the inertial force and moment on a body are deﬁned as the force and moment exerted at the center of mass of the body and whose magnitude is given respectively by the mass of the link times the acceler- ation of the center of mass and the inertial tensor of the link times the angular acceleration of the body. These forces and moments are applied in a direction opposite to the direction of the linear and angular accelerations. As it is well known, introducing these virtual forces and moments in the system allows one to consider it as if it were in “static” equilibrium. To this end, the acceleration of the center of mass and the angular accelerations must ﬁrst be computed. 2.1.1. Inverse Kinematics From the geometry of the mechanism, one can write l1 cos θ + l2 cos α = x0 + l3 cos φ, (1) l1 sin θ + l2 sin α = y0 + l3 sin φ. (2) Eliminating angle α from Equations (1) and (2), one obtains A cos θ + B sin θ = C, (3) where A = x0 + l3 cos φ, B = y0 + l3 sin φ, C = [(x0 + l3 cos φ)2 + (y0 + l3 sin φ)2 + l2 1 − l2 2]/(2l1).
- 3. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 319 The solution of Equation (3) then leads to sin θ = BC + KA √ A2 + B2 , cos θ = AC − KB √ A2 + B2 , where K = ±1 is the branch index, and = A2 + B2 − C2 . (4) Then, from Equations (1) and (2) one can obtain sin α = (x0 + l3 cos φ − l1 cos θ)/l2, cos α = (y0 + l3 sin φ − l1 sin θ)/l2. 2.1.2. Velocity Analysis Differentiating Equations (1) and (2) with respect to time, one has D˙t = e, (5) where D = −l1 sin θ −l2 sin α l1 cos θ l2 cos α , e = −l3 ˙φ sin φ l3 ˙φ cos φ , ˙t = ˙θ ˙α . (6) From Equation (5) one obtains ˙t = D−1 e. (7) 2.1.3. Acceleration Analysis Differentiating Equation (5) with respect to time, one then obtains D¨t = h, (8) where ¨t = ¨θ ¨α , h = ˙e − ˙Dt (9) and the solution of Equation (8) leads to ¨t = D−1 h. (10)
- 4. 320 J. WANG AND C.M. GOSSELIN r C C l l lC m m 1 1 2 2 1 3 3 3 2 0 0 (x , y ) x y O 3 A B O’ r r 1 θ 2 m α φ Figure 1. Geometric representation of the four-bar linkage. Having obtained the angular velocity and acceleration of each link, one can easily compute the acceleration of the centers of mass as a1 = ¨θEr1 − ˙θ2 r1, (11) a2 = ¨θEl1 − ˙θ2 l1 + ¨αEr2 − ˙α2 r2, (12) a3 = ¨φEr3 − ˙φ2 r3, (13) where ri and ai (i = 1, 2, 3) are respectively the position vector and the accel- eration of the center of mass of the ith link, E is a rotation matrix written as E = 0 −1 1 0 , l1 is the position vector from O to A, as represented in Figure 1, and r1 = r1 cos θ r1 sin θ , r2 = r2 cos α r2 sin α , r3 = r3 cos φ r3 sin φ , l1 = l1 cos θ l1 sin θ . (14) The orientation matrix of the ith (i = 1, 2, 3) moving link can be written as Q1 = cos θ − sin θ 0 sin θ cos θ 0 0 0 1 , Q2 = cos α − sin α 0 sin α cos α 0 0 0 1 , Q3 = cos φ − sin φ 0 sin φ cos φ 0 0 0 1 . (15)
- 5. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 321 2 C 0 0(x , y ) x y O 3 τ m1 C2 C1 m2 m3 w w w 1 3f1 2f f3 Figure 2. The inertial force and moment and the gravity forces exerted on each link. Therefore, the forces and moments acting at the center of mass of each mov- ing link include the inertial force and moment and the gravity, as represented in Figure 2. They can be written as fi = −miai + wi, i = 1, 2, 3 mi = −I0i ˙ωi − ωi × (I0iωi), i = 1, 2, 3, where I0i = QiIiQT i , ω1 = 0 0 ˙θ , ω2 = 0 0 ˙α , ω3 = 0 0 ˙φ , (16) and where wi and Ii (i = 1, 2, 3) are respectively the weight and the inertia tensor of ith link about its center of mass. Vectors mi (i = 1, 2, 3) are the inertial torques acting at the center of mass of the links. 2.2. COMPUTATION OF THE VIRTUAL DISPLACEMENTS OF EACH LINK Differentiating Equations (1) and (2), one obtains Aδx = b, (17) where A = l2 sin α −l3 sin φ −l2 cos α l3 cos φ , b = −l1δθ sin θ l1δθ cos θ , δx = δα δφ , (18)
- 6. 322 J. WANG AND C.M. GOSSELIN where δα and δφ are the virtual angular displacements of the links of length l2 and l3 caused by the virtual angular displacement δθ of the input link of length l1. The virtual linear displacement of the center of mass of each link is then com- puted as follows δ1 = δθEr1, δ2 = δθEl1 + δαEr2, δ3 = δφEr3. 2.3. COMPUTATION OF THE GENERALIZED INPUT FORCES OR TORQUES By application of the principle of the virtual work, one can ﬁnally obtain the gener- alized input force, namely, the torque τ to actuate the four-bar linkage, i.e., letting δθ = 1 one has τ = m1 + m2δα + m3δφ + 3 i=1 (fiδi). (19) The simple example presented above has illustrated the general application of the principle of virtual work to the solution of inverse dynamic problems. Now, a general formulation will be proposed for the application of this principle to the dynamic analysis of parallel manipulators, which is the main purpose of this paper. 3. Spatial Six-Degree-of-Freedom Parallel Manipulator The formulation proposed here will now be derived for a six-degree-of-freedom Gough–Stewart platform. However, it should be kept in mind that this formulation can be applied to any type of parallel mechanism. The six-degree-of-freedom manipulator is represented in Figure 3. It consists of a ﬁxed base and a moving platform connected by six extensible legs. Each extensible leg consists of two links and the two links are connected by a prismatic joint. The moving platform is connected to the legs by spherical joints while the lower end of the extensible legs is connected to the base through Hooke joints. By varying the length of the extensible legs, the moving platform can be positioned and oriented arbitrarily with respect to the base of the manipulator. The base coordinate frame, designated as the Oxyz frame is attached to the base of the platform with its Z-axis pointing vertically upward. Similarly, the moving coordinate frame O x y z is attached to the moving platform. The orientation of the moving frame with respect to the ﬁxed frame is described by the rotation matrix Q. The center of the ith Hooke joint is noted Oi while the center of the ith spherical joint is noted Pi.
- 7. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 323 Figure 3. Spatial six-degree-of-freedom parallel mechanism with prismatic actuators (Gough–Stewart platform). If the coordinates of point Pi in the moving reference frame are noted (ai, bi, ci) and if the coordinates of point Oi in the ﬁxed frame are noted (xio, yio, zio), then one has pi = xi yi zi , pi = ai bi ci , for i = 1, . . . , 6, p = x y z , (20) where pi is the position vector of point Pi expressed in the ﬁxed coordinate frame – and whose coordinates are deﬁned as (xi, yi, zi) – pi is the position vector of point Pi expressed in the moving coordinate frame, and p is the position vector of point O expressed in the ﬁxed frame. One can then write pi = p + Qp i, i = 1, . . . , 6, (21) where Q is the rotation matrix corresponding to the orientation of the platform of the manipulator with respect to the base coordinate frame. This rotation matrix can be written, for instance, as a function of three Euler angles. With the Euler angle convention used in the present work, this matrix is written as Q = cφcθ cψ − sφsψ −cφcθ sψ − sφcψ cφsθ sφcθ cψ + cφsψ −sφcθ sψ + cφcψ sφsθ −sθ cψ sθ sψ cθ , (22) where sx denotes the sine of angle x while cx denotes the cosine of angle x.
- 8. 324 J. WANG AND C.M. GOSSELIN α i βi i O Oi Pi rio ρi x y z x y z i i Figure 4. Vector ρi in spherical coordinates. 3.1. INVERSE KINEMATICS The inverse kinematic problem is deﬁned here as the determination of the position and oriention of each link with respect to the base coordinate frame from the given six independent Cartesian coordinates of the platform x, y, z, φ, θ and ψ. This problem is rather straightforward and has been addressed by many authors. One can write pi in terms of the ith leg’s coordinates which are represented in Figure 4. pi = ri0 + ρi, i = 1, . . . , 6 (23) where ri0 and ρi are the vectors from O to Oi and from Oi to Pi respectively, i.e., ρi = ρi cos αi sin βi ρi sin αi sin βi ρi cos βi , ri0 = xi0 yi0 zi0 , i = 1, . . . , 6. (24) Since xi, yi and zi have been obtained from Equation (21), Equation (23) consti- tutes a system of three equations in three unknowns ρi, αi and βi, which can be easily solved as ρi = (xi − xi0)2 + (yi − yi0)2 + (zi − zi0)2, cos βi = (zi − zi0)/ρi, sin βi = 1 − cos2 βi, (0 ≤ βi ≤ π), cos αi = (xi − xi0)/(ρi sin βi), sin αi = (yi − yi0)/(ρi sin βi).
- 9. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 325 Once ρi, αi and βi are known, the position and orientation of the two links of ith leg are completely determined. 3.2. VELOCITY ANALYSIS In this section, the linear and angular velocities of all moving links will be com- puted from the given independent Cartesian velocities of the platform: ˙x, ˙y, ˙z, ωx, ωy and ωz, where the latter three scalar quantities are the components of the angular velocity vector of the platform, ω. One can write ˙pi in terms of the angular velocity vector of the ith leg noted ωi, i.e., ˙pi = ˙ρir + ωi × ρi, i = 1, . . . , 6, (25) where ˙ρir = ˙ρi cos αi sin βi ˙ρi sin αi sin βi ˙ρi cos βi , ρi = ρi cos αi sin βi ρi sin αi sin βi ρi cos βi , ωi = − ˙βi sin αi ˙βi cos αi ˙αi . (26) Equation (25) can be expressed in matrix form as Cipλip = ˙pi, i = 1, . . . , 6, (27) where Cip = cos αi sin βi −ρi sin αi sin βi ρi cos αi cos βi sin αi sin βi −ρi cos αi sin βi ρi sin αi cos βi cos βi 0 −ρi sin βi , λip = ˙ρi ˙αi ˙βi , (28) and Equation (27) is easily solved for λip which leads to the determination of ˙ρi, ˙αi and ˙βi. Once these quantities are known, the computation of the velocities of the bodies of ith leg is straightforward. 3.3. ACCELERATION ANALYSIS The linear and angular accelerations of each of the moving bodies will now be determined from the given Cartesian accelerations of the platform, i.e., ¨x, ¨y, ¨z, ˙ωx, ˙ωy and ˙ωz, where the latter three scalar quantities are the components of the vector of angular acceleration of the platform, ˙ω.
- 10. 326 J. WANG AND C.M. GOSSELIN Differentiating Equation (25) with respect to time, one obtains ¨pi = ¨ρir + ˙ωi × ρi + ωi × (˙ρir + ωi × ρi), i = 1, . . . , 6, (29) where ¨ρir = ¨ρi cos αi sin βi ¨ρi sin αi sin βi ¨ρi cos αi , ˙ωi = − ¨βi sin αi − ˙βi ˙αi cos αi ¨βi cos αi − ˙βi ˙αi sin αi ¨αi . (30) Equation (29) can be rewritten in matrix form as Cip ˙λip = hi, i = 1, . . . , 6, (31) where Cip is given in Equation (28) and where ˙λip = [ ¨ρi ¨αi ¨βi ]T , hi = ¨xi − 2 ˙ρi ˙βicαi cβi + 2 ˙ρi ˙αisαi sβi + 2ρi ˙αi ˙βisαi cβi + ρi ˙α2 i cαi sβi + ρi ˙β2 i cαi sβi ¨yi − 2 ˙ρi ˙βisαi cβi − 2 ˙ρi ˙αicαi sβi − 2ρi ˙αi ˙βicαi cβi + ρi ˙α2 i sαi sβi + ρi ˙β2 i sαi sβi ¨zi + cβi + 2 ˙ρi ˙βisβi + ρi ˙β2 i cβi , where sx denotes the sine of angle x while cx denotes the cosine of angle x. Equation (31) is readily solved for ˙λip. Once the acceleration components are known, the accelerations of the leg bodies are easily determined. 3.4. GENERALIZED INPUT FORCES The generalized input forces will now be determined by ﬁrst including the iner- tial forces and moments in the system and considering it as if it were in “static” equilibrium. The principle of virtual work will be applied. 3.4.1. Computation of the Force and Torque Acting on the Center of Mass of Each Link According to d’Alembert’s principle, the force acting on the center of mass of each link consists of two parts: the inertial force and the gravity force. Similarly, the moment acting on each link is the inertial moment. In order to compute inertial forces, one must ﬁrst determine the acceleration of the center of mass of each link. One can write aiu = ¨pi + ˙ωi × Qiriu + ωi × (ωi × Qiriu), i = 1, . . . , 6, (32) ail = ˙ωi × Qiril + ωi × (ωi × Qiril), i = 1, . . . , 6, (33) where aiu and ail are respectively the acceleration of the center of mass of the upper and lower links of the ith leg. They are expressed in the ﬁxed coordinate system.
- 11. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 327 Vector riu and ril are the position vectors of the center of mass of the upper and lower links of the ith leg in the local frame, and one would generally have riu = 0 0 riu , ril = 0 0 ril . (34) Moreover, matrix Qi is the rotation matrix from the ﬁxed frame to the local frame attached to the ith leg. The acceleration of the center of mass of the platform can be computed as follows ap = ¨p + ˙ω × Qrp + ω × (ω × Qrp), (35) where ap is the acceleration of the center of mass of the platform and rp is the po- sition vector of the center of mass of the platform, expressed in the frame attached to the platform. Then, the force and moment acting on the center of mass of each link can be directly computed as follows fiu = −miuaiu + wiu, i = 1, . . . , 6, (36) fil = −milail + wil, i = 1, . . . , 6, (37) miu = −I0iu ˙ωi − ωi × (I0iuωi), i = 1, . . . , 6, (38) mil = −I0il ˙ωi − ωi × (I0ilωi), i = 1, . . . , 6, (39) where I0iu = QiIiuQT i , I0il = QiIilQT i , (40) and where fiu, miu, fil and mil denote the force acting on the upper link, the mo- ment acting on the upper link, the force acting on the lower link and the moment acting on the lower link of the ith leg. Iiu and Iil are the inertia tensor computed in body reference frame of the upper and lower links of the ith leg respectively and miu and mil are their masses. Vectors wiu and wil are the weight vectors, i.e., wiu = 0 0 −miug , wil = 0 0 −milg . (41) Finally, one has fp = −mpap + wp, (42) mp = −I0p ˙ωp − ωp × (I0pωp), (43) where I0p = QIpQT , (44)
- 12. 328 J. WANG AND C.M. GOSSELIN 6u o o o 1 2 4 3 x’ z’ y’ 3 2 o4 2 1 3 4 1 O’ platformP P P P P 6 1u 2u 3u 4u1l 2l 3l 4l p p 1u 2u 2l 1l 4l 4u 3u P 5u 5u 5l 5l 6 6 5 7 6u 6l 6l x y z m m m m m m m m m m m m 3l O f f f f f f f f f f f f m f Figure 5. Inertial forces and moments on each of the links of the system. and where fp and mp denote the force and moment acting on the platform and where Ip, mp and wp are the inertia tensor computed in body reference frame, the mass and the weight vector of the platform, respectively. The inertial forces and moments acting on the center of mass of each link of the manipulator are represented schematically in Figure 5. 3.4.2. Computation of the Virtual Displacements of the Links From Equation (23) one has ρi = pi − ri0, i = 1, . . . , 6. (45) Taking the square of the norm of Equation (45) leads to ρ2 i = (xi − xio)2 + (yi − yio)2 + (zi − zio)2 , i = 1, . . . , 6. (46) Differentiating both sides of Equation (46) with respect to time, one then obtains ρi ˙ρi = (xi − xio)˙xi + (yi − yio) ˙yi + (zi − zio)˙zi, i = 1, . . . , 6, (47) where ˙xi, ˙yi and ˙zi are the three components of the velocity vector ˙pi of point Pi. They can be computed using the following equation: ˙pi = ˙p + ω × Qpi, i = 1, . . . , 6, (48) where ˙p and ω are respectively the velocity of point O and the angular velocity of the platform, i.e., ˙p = ˙x ˙y ˙z , ω = ωx ωy ωz . (49)
- 13. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 329 Substituting Equation (48) into Equation (47) and writing it in matrix form, one then obtains B˙ρ = A˙x, (50) where ˙ρ = [ ˙ρ1 ˙ρ2 ˙ρ3 ˙ρ4 ˙ρ5 ˙ρ6 ]T , ˙x = [ ˙x ˙y ˙z ωx ωy ωz ]T , and where A and B are Jacobian matrices written as B = ρ1 0 0 0 0 0 0 ρ2 0 0 0 0 0 0 ρ3 0 0 0 0 0 0 ρ4 0 0 0 0 0 0 ρ5 0 0 0 0 0 0 ρ6 . (51) and A = ρ1x ρ1y ρ1z (b1ρ1z − c1ρ1y) (c1ρ1x − a1ρ1z) (a1ρ1y − b1ρ1x) ρ2x ρ2y ρ2z (b2ρ2z − c2ρ2y) (c2ρ2x − a2ρ2z) (a2ρ2y − b2ρ2x) ρ3x ρ3y ρ3z (b3ρ3z − c3ρ3y) (c3ρ3x − a3ρ3z) (a3ρ3y − b3ρ3x) ρ4x ρ4y ρ4z (b4ρ4z − c4ρ4y) (c4ρ4x − a4ρ4z) (a4ρ4y − b4ρ4x) ρ5x ρ5y ρ5z (b5ρ5z − c5ρ5y) (c5ρ5x − a5ρ5z) (a5ρ5y − b5ρ5x) ρ6x ρ6y ρ6z (b6ρ6z − c6ρ6y) (c6ρ6x − a6ρ6z) (a6ρ6y − b6ρ6x) , (52) where ρix = xi − xio, ρiy = yi − yio, ρiz = zi − zio, ai = q11ai + q12bi + q13ci, bi = q21ai + q22bi + q23ci, ci = q31ai + q32bi + q33ci, i = 1, . . . , 6, in which qij (i, j = 1, 2, 3) is the ith row and jth column element of matrix Q. Let δα j i and δβ j i be the virtual angular displacements of the ith leg associated with jth actuated joint corresponding to angles αi and βi (i, j = 1, . . . , 6), let δxj , δyj , δzj and δϕj x, δϕj y, δϕj z be the virtual displacements of point O and the virtual angular displacements of the platform associated with jth actuated joint and let δρj be the virtual displacement of the jth actuated joint.
- 14. 330 J. WANG AND C.M. GOSSELIN Using Equation (50), one can compute the linear and angular virtual displace- ments of the platform associated to the virtual displacement of the jth actuated joint, i.e., δxj = A−1 Bδρj , j = 1, . . . , 6, (53) where δxj = [ δxj δyj δzj δϕ j x δϕ j y δϕ j z ] , j = 1, . . . , 6, δρ1 = [ 1 0 0 0 0 0 ]T , δρ2 = [ 0 1 0 0 0 0 ]T , δρ3 = [ 0 0 1 0 0 0 ]T , δρ4 = [ 0 0 0 1 0 0 ]T , δρ5 = [ 0 0 0 0 1 0 ]T , δρ6 = [ 0 0 0 0 0 1 ]T . Having obtained the virtual displacements of the platform of the manipulator, the virtual displacements of ith leg associated with the jth actuator can be easily obtained from Equation (27), i.e., δλ j ip = C−1 ip δp j i , i, j = 1, . . . , 6, (54) where δλ j ip = [ δρj δα j i δβ j i ] , i, j = 1, . . . , 6, (55) and δp j i is the virtual displacement of point Pi associated with a unit virtual dis- placement of the jth actuator. This virtual displacement can be computed from the virtual displacements of the platform, i.e., δp j i = δpj + δϕj × Qpi , i, j = 1, . . . , 6, (56) where δpj = δxj δyj δzj , δϕj = δϕ j x δϕ j y δϕ j z . (57) Once the virtual linear and angular displacements of each link of the manipula- tor are known, the virtual displacements of the center of mass of each link can be computed as follows δ j iu = δp j i + δϕ j i × Qiriu, i, j = 1, . . . , 6, δ j il = δϕ j i × Qiril, i, j = 1, . . . , 6, δj p = δpj + δϕj × Qrp, i = 1, . . . , 6,
- 15. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 331 where δ j iu, δ j il and δj p are the virtual displacements of the center of mass of the links of the ith leg associated with a virtual displacement of the jth actuator, and where δϕ j i = −δβ j i sin α j i δβ j i cos α j i δα j i , i, j = 1, . . . , 6. (58) 3.4.3. Computation of Actuated Force/Torque Using the principle of virtual work and letting τi (i = 1, . . . , 6) be the actuating force of ith actuated joint, one then has τj = fpδj p + mpδϕj + 6 i=1 [fiuδ j iu + filδ j il + (miu + mil)δϕ j i ], j = 1, . . . , 6 which thereby completes the procedure. 4. Example In this section, an example is given in order to illustrate the results. The prescribed trajectory is one in which the platform of the manipulator translates along a simple sine trajectory while maintaining a ﬁxed orientation. The parameters used in this example are given as xo1 = −2.120, yo1 = 1.374, xo2 = −2.380, yo2 = 1.224, xo3 = −2.380, yo3 = −1.224, xo4 = −2.120, yo4 = −1.374, xo5 = 0.0, yo5 = 0.15, xo6 = 0.0, yo6 = −0.15, zoi = 0.0, (i = 1, . . . , 6), a1 = 0.170, b1 = 0.595, c1 = −0.8, a2 = −0.6, b2 = 0.15, c2 = −0.8, a3 = −0.6, b3 = −0.15, c3 = −0.8, a4 = 0.170, b4 = −0.595, c4 = −0.8, a5 = 0.430, b5 = −0.845, c5 = −0.8, a6 = 0.430, b6 = 0.445, c6 = −0.8, l5 = 1.5, ρi max = 4.5, ρi min = 0.5, (i = 1, . . . , 6), mp = 1.5, miu = mil = 0.1, riu = ril = 0.5, (i = 1, . . . , 6), rp = 0, Iiu = Iil = I6 = 1/160 0 0 0 1/160 0 0 0 1/1600 , (i = 1, . . . , 6), Ip = 0.08 0 0 0 0.08 0 0 0 0.08 ,
- 16. 332 J. WANG AND C.M. GOSSELIN çघNङ !tघradङ ç2 ç1 1 2 3 4 5 6 6 8 10 12 Figure 6. Input force at actuated joints 1 and 2. çघNङ !tघradङ ç3 ç4 1 2 3 4 5 6 6.5 7.5 8 8.5 Figure 7. Input force at actuated joints 3 and 4. where the lengths are given in meters, the masses in kilograms and the inertias in kilograms meter square. The speciﬁed trajectory can be expressed as x = −1.5 + 0.2 sin ωt, y = 0.2 sin ωt, z = 1.0 + 0.2 sin ωt, φ = 0, θ = 0, ψ = 0, where ω = 3.0, (0 ≤ ωt ≤ 2π). The generalized input forces obtained at the six actuated joints are represented in Figures 6–8. This example has been veriﬁed by the classical Newton–Euler approach. The two approaches lead to identical results and the approach based on the principle of virtual work leads to a faster algorithm which is about 30% faster than the one obtained using the Newton–Euler approach.
- 17. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 333 çघNङ !tघradङ ç5 ç6 1 2 3 4 5 6 6 7 8 9 10 11 12 Figure 8. Input force at actuated joints 5 and 6. 5. Conclusion A new approach for the dynamic analysis of parallel manipulators has been pro- posed in this paper. This approach is based on the well known principle of virtual work. The principle of virtual work has ﬁrst been recalled and illustrated through the dynamic analysis of the four-bar linkage. Then, the dynamic analysis of spa- tial six-degree-of-freedom parallel manipulators with prismatic actuators has been performed. The procedure described here leads to efﬁcient algorithms and can be applied to any type of parallel mechanism. Acknowledgment The authors would like to acknowledge the ﬁnancial support of the Natural Sci- ences and Engineering Research Council of Canada (NSERC) as well as the Fonds pour la Formation de Chercheurs et l’Aide à la Recherche du Québec (FCAR). References 1. Angeles, J., Rational Kinematics, Springer Tracts in Natural Philosophy, Vol. 34, Springer- Verlag, Berlin/New York, 1988, 65–66. 2. Blajer, W., Schiehlen, W. and Schirm, W., ‘Dynamic analysis of constrained multibody systems using inverse kinematics’, Mechanism and Machine Theory 28(3), 1993, 397–405. 3. Craver, W.M., ‘Structural analysis and design of a three-degree-of-freedom robotic shoulder module’, Master Thesis, The University of Texas at Austin, 1989. 4. Do, W.Q.D. and Yang, D.C.H., ‘Inverse dynamic analysis and simulation of a platform type of robot’, Journal of Robotic Systems 5(3), 1988, 209–227. 5. Fichter, E.F., ‘A Stewart platform-based manipulator: general theory and practical construc- tion’, The International Journal of Robotics Research 5(2), 1986, 157–182. 6. Gosselin, C., ‘Determination of the workspace of 6-DOF parallel manipulators’, ASME Journal of Mechanical Design 112(3), 1990, 331–336.
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