2.1. Notation

As shown in Figure 2, this hexapod system consists of a moving platform MP to which a tool is attached, and six legs sliding along the guideways that are mounted on the support structure including the base platform BP. Each leg is connected at one end to the guideway by a universal joint and at another end to the moving platform by a spherical joint.

Figure 2.

Kinematic notation of the ith leg

The coordinate systems used are a fixed coordinate system O-xyz is attached to the base and a local coordinate system O_t-x _t y _t z _t attached to the moving platform. Vector b_i, s_i, and l_i are directed from O to B_i, from B_i to U_i, and from U_i to S_i respectively. B_i indicates the position of one end of the ith guideway attached to the base, U_i indicates the position of the ith universal joint, and S _i indicates the position of the ith spherical joint. Six legs are numbered from 1 to 6.

Furthermore, a local coordinate frame O_i-x_iy_iz_i is defined for each leg, with its origin located at the center of the ith universal joint. Two unit vectors are used. Unit vector uil is along the leg length representing the direction of the ith leg, and unit vector uis is along the guideway representing the direction of the ith guideway. The orientation of the ith coordinate frame with respect to the base can therefore be defined by a 3×3 rotation matrix, for i = 1,…,6, as

where uia is expressed as

Note that vector uil is configuration-dependent and determined for the given location of the moving platform; vector uis​ is constant and defined by the geometry of the hexapod.

For the purpose of carrying out the inverse dynamics analysis of the hexapod, the following symbols are defined. As shown in Figure 2, C_i is the center of mass of the ith leg, C_p is the center of mass of the moving platform, ci c˙i and c¨i are the position, velocity and acceleration vectors, respectively, of C_i with respect to the fixed coordinate frame, ρ¯​ is the vector pointing from O_t to C_p with respect to the local coordinate frame O_t-x_ty_tz_t.

2.2. Kinematics

Consider one branch of the leg-guideway system, as shown in Figure 2, the following loop equation for i = 1,…,6, holds,

where h and R are the vector and rotation matrix that define the position and orientation of the moving platform relative to the base, respectively, p¯i is the vector representing the position of the ith spherical joint on the moving platform in the local coordinates.

Since the leg always moves along the guideway, s_i can be expressed as

where si is a scalar representing the displacement of the ith actuator along the guideway. Likewise, leg vector li can be expressed as

where l _i is a scalar representing the fixed length of the ith leg. As mentioned in Section 2.1, the leg axis is parallel to the z_i axis of the local coordinate frame O_i-x_iy_iz_i. In the light of eq.(1), uil can be expressed as

Substituting eqs.(4 & 5) into eq.(3) and rearranging it yields the following kinematics equations for the fixed-length leg hexapod, for i = 1,…,6,

To obtain the velocity of the moving platform, taking the time derivative of eq. (7) yields

where v and ω are the vectors representing the velocity and angular velocity of the moving platform, respectively, and ωi is the vector representing the angular velocity of the ith leg.

Furthermore, by taking dot product on both sides of eq.(8) by l_i, it leads to

It is well known that the kinematic analysis of parallel manipulator leads to two Jacobian matrices, namely, the forward and the inverse Jacobian (Gosselin & Angeles, 1990). To find the Jacobians for the hexapod under study, rearranging eq.(9) yields the following form

where tp=[vT,ωT]T is the 6×1 twist vector of the moving platform. Consider all six legs it leads to the following expression

where s˙=[s˙1,...,s˙6] T is the 6×1 vector of the actuator speeds, and A and B are the 6×6 matrices representing the inverse and forward Jacobian of the hexapod and they are defined as

Eq.(11) defines the differential relationship between the actuator speeds s˙ and the twist of the moving platform t_p. Rewriting eq.(11) gives

Provided that B is invertible, the Jacobian matrix of the moving platform Jp can be given as

where

for i = 1,..,6. From eq.(14), t_p can be expressed in terms of s˙ as,

where

To obtain the acceleration of the moving platform, taking the time derivative of eq. (14) yields

where s¨=[s¨1,...,s¨6] T is the 6×1 vector of the actuator accelerations, t˙p=[aT ω˙T]T is the time derivative of the twist of the moving platform, J˙p is the time derivative of the Jacobian matrix of the moving platform obtained by differentiating Jp with respect to time, that is

where A˙ and B˙ given as

If the mass of the leg is uniformly distributed, then the center of mass is in its middle. The velocity of the center of mass can be given as

Upon differentiating eq.(22), the acceleration of the center of mass can be given as

To obtain the leg angular velocity and acceleration, denote by E_i the 3×3 cross-product matrix associated with vector uil , then eq.(9) may be re-written as

Consider all six legs, it forms a set of linear equations containing the unknowns of the leg angular velocity. There are three components of ωi for each leg. Because matrix Ei is a skew symmetric and singular, it is impossible to directly solve eq.(24). However, since the leg does not spin about its longitudinal axis, this indicates (Tsai, 2000)

In the light of eq.(25), eq. (24) may be rewritten as

where Ai is a 4×3 matrix and ei is a 4-dimensional vector, and they are defined as

Solving eq. (26) leads to the expression for the leg angular velocity

Now eq.(29) is substituted back into eq.(22), and the velocity becomes

By examining eqs.(29) and (30), it may be noted that the two terms in the brackets are identical. The first term may be expressed as

where Epi is the cross-product matrix of Rp¯i . In the light of eq.(17), eq.(31) may be related to s˙ as

where 1 is the 3×3 identity matrix and T1i is the 3×6 matrix pertaining to the first term defined as

The second term in eqs.(29) and (30) can also be expressed in terms of s˙

where T2i is the 3×6 matrix pertaining to the second term defined as

In eq.(35), 0₃ is the 3-dimensional null vector. The twist of the ith leg can be expressed in terms of s˙ as

where t_i is the twist of the ith leg, i.e. ti=[c˙iT,ωiT]T , and the 6×6 matrix T_i is given as

Furthermore, the leg angular acceleration can be obtained by differentiating eq.(26) with respect to time, that is

where

From eq.(38), vector ω˙i representing the angular acceleration of the ith leg is given as

3.1. The natural orthogonal complement method

Prior to performing dynamic modeling of the hexapod, a brief review of the natural orthogonal complement method (Angeles & Lee, 1988) is provided. Consider a system composed of p rigid bodies under holonomic constraints, the Newton-Euler equations for each individual body can be written, for i = 1,.., p, as

where ti is the twist of the ith body, wi=[niT,fiT]T represent the wrench acting on the ith body, n_i and f_i are the resultant moment and the resultant force acting at the center of mass. In general w_i can be decomposed into working wrench wiw and non-working wrench wiN The former can further be decomposed as

where wia,wig and wid are the actuator, gravity and dissipate wrenches, respectively.

In eq. (42), the 6×6 angular velocity matrix W_i and the 6×6 inertia matrix M_i are defined as

where I_i is the 3×3 matrix of the moment of inertia of the ith body, m_i is the body mass, O denotes the 3×3 null matrix, and e is an arbitrary vector.

If consider all p bodies, the assembled system dynamics equations are given as

where the 6p×6p generalized mass matrix M and generalized angular velocity matrix W are defined as

and the 6p-dimensional generalized twist t, generalized working wrench wW and generalized non-working wrench wN are defined as

It can be shown that the kinematic constraints hold the following relation with the generalized twist

where 06p is the 6p-dimensional null vector, K is the 6p×6p velocity constraint matrix with a rank of m which is equal to the number of independent holonomic constraints. The number of degrees of freedom of the system, i.e. independent variables, is determined as n = 6p - m. Denote the independent variables by s, they can be related to the twist as

where T is a 6p×n twist-mapping matrix.

By substituting eq.(51) into eq.(50), the following relation can be obtained

where T is the natural orthogonal complement of K. As shown in (Angeles & Lee, 1988, 1989) the non-working vector w lies in the null space of the transpose of T. Thus, if both sides of eq. (46) are multiplied by T, in the aid of eqs. (51 & 52), the system dynamics equations can be obtained as

where the n×n generalized inertia matrix I and coupling matrix C are defined as

Furthermore, by defining the following generalized forces

the inverse dynamics of the system can be given as

where τa is the vector representing the applied actuator forces.

3.2. Inverse dynamics

The key in applying the natural orthogonal complement method is to derive the expression for the twist-mapping matrix T, which relates the speeds of the independent variables to the generalized twist. For the hexapod under study, the independent variable s is the vector representing the actuator displacement, with the total number of six, as defined before. The generalized twist is expressed as

Note that t₁ to t₆ are the twists for the six legs. Since the twist in eq.(36) is defined at the center of mass of the leg, T_i represents the twist-mapping for the legs. For the moving platform, t_pc is defined as the center of mass which may be expressed as

Differentiating eq.(59) gives

In the light of eq.(60), the following relation can be obtained

t_pc = H_p t_p

where Hp is the 6×6 matrix defined as

In eq.(62), E is the cross-product matrix of R​​ρ¯ . Note that when ​ρ¯ is zero, i.e. the center of mass coincides with the coordinate origin, H_p becomes an identity matrix, and t_pc = t_p.

The twist-mapping matrix T for the hexapod under study can be given in the light of eqs.(17), (36) and (62) as

where T is a 42×6 matrix. With T, the generalized forces can be defined according to eq.(56), and the applied actuator forces can be determined according to eq.(57).

4.1. Geometric and inertial parameters

The geometry of the base and the moving platform is shown in Figure 3.

Figure 3.

Geometry of the base and the moving platform

Accordingly, the coordinates of vector b_i with respect to the fixed frame are given as

where L_b and l_b are the long and short side of the base hexagon, cs=cos(30∘) , and yb=(Lb/2+lb)tg(30∘) . Likewise, the coordinates of vector ​p¯i with respect to the local frame are given as

where L_p and l_p are the long and short side of the moving platform hexagon, and yp=(Lp/2+lp)tg(30∘) . The geometric parameters and inertial parameters are given in Tables 1 and 2, respectively. In Table 1, S is the guideway length, is the guideway angle between the guideway and the vertical direction, and l is the length of the leg. These three parameters are the same for all the guideways and legs. Parameters L_b, l_b, L_p and l_p are defined in Figure 3. In Table 2, m is the mass, and I_xx, I_yy and I_zz are the moments of inertia.

4.2. Numerical example

A simulation program has been developed using Matlab based on the method described in the previous sections. In terms of computation, as can be seen from eq. (54), the inverse dynamics of the hexapods mainly involves the twist-mapping matrix T and its derivative, which could be computed numerically for each time interval. This way, it is computationally more efficient. To further speed up computation, parallel computation techniques could be used. As shown in Figure 4, the motion part including actuator speeds and accelerations could be computed in parallel to the inertia part including mass matrix I and coupling matrix C. Since the program is done in Matlab, parallel computation is not realized. However, this strategy can certinaly be applied to model-based control using the dynamic equations.

In terms of singularity, as can be shown from eq. (63), the twist-mapping matrix T becomes degenerate when the moving platform Jacobian J_p (T_p = J_p ) is singular.

The movement of the moving platform is defined in terms of 3-4-5 polynomials that guarantee zero velocities and zero accelerations at the beginning and at the end. The selection of a smooth motion profile is very important for the hexapod as it is operated under high speeds. The conventional machine tools are run at a maximum velocity of 30m/min with a maximum acceleration of 0.3 g. Hexapods can run at a maximum velocity of 100 m/min with a maximum acceleration over 1 g.

The first simulation is for high speed, with a maximum velocity of 102 m/min. The second simulation is for low high speed with a maximum velocity of 30 m/min. In both cases, the hexapod moves a distance of 0.1m along the z axis. The initial position of the moving platform is at x_o = 0, y_o = 0 and z_o = 0.7m. Figure 4 shows the velocity profiles of the moving platform. Figures 5, 6 and 7 show the displacements, velocities and accelerations of the six actuators, respectively. Figure 8 shows the computed actuator forces. The simulations show that high speed motions result in large actuator forces.

Figure 4.

Motion Profile of moving platform

Figure 5.

Actuator displacements

Figure 6.

Actuator velocities

Figure 7.

Actuator accelerations

5.1. Static balancing using counterweight

The static balancing of the parallel manipulator under study is investigated using counterweights. The base coordinate frame Oxyz frame, is fixed to the base with Z-axis pointing vertically upward and the moving coordinate frame O'x'y'z' is attached to the moving platform. The Cartesian coordinates used to describe the pose of the platform are as shown in Fig. 9 given by the position of O' with respect to the fixed frame and the orientation of the platform represented by the rotation matrix Q

Using the counterweights, static balancing is obtained if the global center mass of the mechanism is kept stationary at any values of the independent variables. To choose an suitable constant, namely

Figure 9.

Kinematic mode

where r is the position vector of the global mass center, and M is:

where mp is the mass of the platform, mi is the mass of the leg. The global centre of the mass of the manipulator is written as

where rp is the platform center of the mass, ri is the leg center of the mass. From Fig. 9, vectors rp , ri can be derived, and substituted into eq.(69), yielding

where g is the vector center of mass of the moving platform with respect to the frame O'x'y'z', h is the position of O' with respect to the fixed frame, p_i is the position of the spherical joint with respect to the moving coordinate frame, b_i is the position of the lower end of the guideway with respect to the fixed frame, l_i is the length of the leg, s_i can be written, for i=1,…,6, as

where s^i is the unit vector of guideway, ρ_i is the independent variable of the prismatic joint. In concise form, eq. (70) is expressed as

where

The conditions for static balancing can be given for i=1,..,6, as follows:

From conditions A5i=0 i=1,..,6, one can obtain

By condition A1=0 , one can obtain

Eq.(79) shows that the balancing by counterweight is impossible. If it was substituted in the condition B= 0,

then one can obtain

From eq. (81), it shows that the manipulator could be balanced by a device that provide a force that is

equal to the weight of the links and the platform;

in opposite direction of the weight;

5.2. Static balancing wit a pantograph counterweight

Since it is shown that the static balancing of the examined mechanism is impossible with the help of counterweights, we propose a method to add a pantograph connecting the moving platform O' to the fixed platform O, as shown in Fig. 10. The pantograph is a device that allows to keep two end points on the same line and keep their distance at the centre with a constant ratio. In this application it is possible to use a pantographs with two or more mesh as shown in Fig. 10 and Fig. 11, respectively. In both case the manipulator is balanced. The pantograph is fixed to the moving platform on the point O' by a spherical joint and fixed to the point O by an universal joint. The leg counterweight is shown in Fig.12.

Figure 10.

Model with counterweights mass

Figure 11.

Balanced Hexapod using pantograph

Figure 12.

Leg counterweight

In this case, the mass M becomes,

where mp and mp∗ are the mass of the platform and the mass of the platform counterweight, mi and mi∗ are the mass of the legs and the mass of the legs counterweights, ma and ma∗ are the mass of the pantograph and the mass of the of the pantograph counterweight. In this case, the global center of the mass of the manipulator is written as

where rp and rp∗ are the platform center of the mass and the platform counterweight position, ri and ri∗ are the legs center of the mass and the legs counterweight position, ra and ra∗ are the pantograph center of the mass and the pantograph counterweight position. From Figs. 9-10, vectors rp , rp∗ , ra ra∗ , ri and ri∗ can be derived and substituted into eq.(83), yielding

where, l_a is the center of mass of the pantograph with respect to the fixed frame, l_a is the pantograph counterweight position with respect to the fixed frame, l_gi is the length of the leg counterweight link, l_i is the length of the leg, s_i can be written, for i=1,...,6, as

In concise form, eq.(84) can be expressed as

where

The conditions for static balancing can be given, for i =1,..,6, as follows

From conditions A5i=0 , for i=1,..,6, one can obtain

From eq. (92), for i=1,..,6, the following is obtained

By condition A1=0 , i.e.,

one can obtain

Finally, condition B= 0 leads to the following

Eq.(96) shows that the static balancing can be achieved by fixing the global center of the mass of the moving platform, that of the legs and their counterweights at the same position, O'. In order to obtain it, the platform counterweight should be placed in the position:

Simulation is carried out to demonstrate the proposed method. The results are shown in Figs. 13-14, from which it can be seen that the centre of mass of the robot is non-stationary for non balanced case, while it is fixed for the balanced case.

After static balancing the global mass of the device increases by

The negative effect for the dynamic performance by the increasing global mass can be reduced by optimum design of the pantograph. A graph can be arranged to provide such help. Fig. 15 shows the ratios,

which vary respect to the ratio r_a / h and l_gi / l_gi and where I_i is the moment of inertia of the leg, I_i is the moment of inertia of the leg counterweight whit respect of P_i, I_a is the moment of inertia of the moving platform and I_a is the moment of inertia of the pantograph counterweight with respect of O. It should be noted that with a suitable design it is possible to reduce ΔM at the same time, it may increase I_i and I_a. The effect of gravity compensation on the dynamic performances was studied in detail in (Xi, 1999).

Figure 13.

Mobile center of mass Hexapod

Figure 14.

Fixed center of mass of Balanced Hexapod

Figure 15.

Graph for optimum design