Finding the Generalized Forces of a Series-Parallel Manipulator

Abstract

In this work the kinematic and dynamic analyses of a robot manipulator whose topology consists of parallel kinematic structures with linear actuators are approached by means of the theory of screws and the principle of virtual work. The input/output equations of velocity and acceleration are obtained by applying screw theory. Then the generalized forces of the manipulator are determined combining screw theory and the principle of virtual work. Finally, a case study, whose numerical results are compared with simulations generated with the aid of specialized software, is included.

Keywords

Parallel manipulator Virtual work Klein form Screw theory Dynamics

1. Introduction

The kinetic analysis of a robot is a crucial step in its design process. The computation of the required forces/torques associated to the generalized coordinates of the robot, taking into proper account the inertial and mass properties of the corresponding moving bodies, allows selecting the servo motors and determining the dimensions of the links. In that way, over recent years much research has been conducted on the dynamics of robots with parallel kinematic structures, for instance [1]–[8].

The methods of Newton-Euler and the Lagrangian are the most used to perform the dynamic analysis of parallel mechanisms [3, 4, 11, 12]. Other authors solve this with the formulations of Kapel, Routh and Apell equations. The Newton-Euler equations consider reactions of kinematic pairs, which do not exclude the explicit formulation between movements and forces, creating a large number of equations. Moreover, the Lagrangian, Routh and Apell methods involve systems of nonlinear differential equations that are computationally less efficient.

Currently, the more popular approach to achieving the dynamic model of parallel robots is the use of the virtual work principle [9 –12, 15 –18], which avoids the calculation of the internal reactions in the mechanism. In addition, the combination of this principle and screw theory significantly simplifies the model [17,18].

In Gallardo-Alvarado et al.[19] a manipulator whose topology is based upon parallel kinematic structures was proposed, among other applications, as a multi-axis machine tool. However, the kinetic analysis was not considered in that contribution. Hence, in this work the generalized forces of that robot are obtained by applying the principle of virtual work and screw theory.

2. Robot architecture

In recent years much effort has been made to introduce robots with parallel kinematic structures for applications like milling, grinding, deburring, laser cutting and welding, high precision assembly, medical devices, etc. However, due to their multi-closed-chain topology, the forward displacement analysis (FDA) of most parallel manipulators does not allow closed-form solutions, e.g., the FDA of the general Gough-Stewart platform [20, 21, 22] leads to a forty-order univariate polynomial equation [23] which leads to difficultyin determining the actual configuration of the hexapod. Thus, due to the lack of closed-form solutions, complementary methods, such as fine adjustment of robot positions, on site servo tuning, kinematic error identification and compensation, iterative learning control with external measurement systems, 2D and 3D vision-based trajectory adjustment, force control and fuzzy logic, are used to maintain the successful performance of the robot. Series-parallel and hybrid manipulators bring the opportunity to obtain semi-closed form solutions of the FDA preserving most of the advantages of parallel manipulators. Along those lines, Gallardo-Alvarado et al. [19] introduce a manipulator based on parallel kinematic structures called LinceJJP.

As shown in Figure 1, the LinceJJP-type manipulator consists of a fixed platform (A), a middle platform (B) and an output platform (C). The bodies A and B are connected to each other through three RPS type limbs, numbered from 1 to 3, where the prismatic joints play the role of active kinematic pairs, having associated generalized coordinates q̄_i, unless other values are indicated in the remainder of the contribution i = 1,2,3. Hereafter R, P, S, and U stand for revolute, prismatic, spherical and universal joints, respectively. Thus, the middle platform can realize two rotations and one translation with respect to the fixed platform given the limb lengthsd_i.

Figure 1.

Manipulator LinceJJP

The middle and output platforms are connected to each other by means of three passive RPS type kinematic chains that constrains one rotation between bodies C and B. Furthermore, three UPS type kinematic chains connect the fixed and output platforms where the prismatic joints are selected as active elements of the robot having associated generalized coordinates q _i ., It is evident that according to the UPS type limbs the output platform can reach arbitrary poses with respect to the fixed platform, however, the connection of body C to body A through two 3-RPS parallel manipulators indicates that the orientation of body C with respect to body A cannot be arbitrary because there is a lost rotation between the output and fixed platforms.

3. Velocity and acceleration analysis

The mathematical tools used to approach the dynamic analysis of the robot are the theory of screws and the principle of virtual work, to this end the modelling of the screws are shown in Fig. 2.

Figure 2.

Infinitesimal screws of the manipulator

Before we proceed, it is necessary to emphasize that in order to satisfy an algebraic requirement, the revolute joints of the robot are modelled as cylindrical joints in which the corresponding translational velocities vanish, in other words 0ωⁱ₁ = 0ωⁱ₇ = 0. of course this consideration does not affect the mobility of the robot. On the other hand, the Klein form {*; *} of the Lie algebra e(3) plays a central role in the kinematic and dynamic analysis of parallel manipulators.

An infinitesimal screw $, or twist, is a vector in $ ∈ ℝ⁶ that consists of a primal part, P ($)=s, and a dual part, D($) = so, where s is a unit vector along the instantaneous axis of the screw; s_O is the moment produced by vector s with respect to a vector r_O/P:

s_{O} = h s + s \times r_{O ∕ P},

where h is the screw pitch.

Given two screws$₁ = (s₁ s_O₁) and $₂ = (s₂, s_O₂) of e(3), the Klein form is defined as

{$_{1}; $_{2}} = s_{1} • s_{O 2} + s_{2} • s_{O 1},

(1)

where the dot. denotes the inner product of the usual three-dimensional vectorial algebra. It is said that the screws$₁ and $₂ are reciprocal if {$₁; $₂} = 0.

With reference to Figs. 3a and 3b, if the screws $₁ and $₂ represent revolute joints, then these screws are reciprocal when their primal partsintersect or are parallel. Another possibility of reciprocal screws immediately emerges if the primal part of $₁, representing a revolute joint, is perpendicular to the dual part of the screw $₂denotinga prismatic joint, see Fig. 3c.

Figure 3.

Typical reciprocal screws

3.1 Velocity analysis

The velocity state, or twist about a screw, of a body 6 with respect to another body or reference frame a is defined as a six-dimensional vector given by

​^{a} V ​_{O}^{b} \equiv [\begin{matrix} ​^{a} ω ​^{b} \\ ​^{a} v ​_{O}^{b} \end{matrix}],

(2)

where ^aω^b and ^av^b_o are, respectively, the angular and linear velocities of a point O of body b, which is instantaneously coincident with a point embedded in body a. Usually, point O is the origin of the reference frame attached at body a. On the other hand, assuming that P is a point of interest of body 6, for example its centre of mass, then the velocity state ^a V ^b _P can be obtained applying the concept of helicoidal vector field [24] as

​^{a} V ​_{P}^{b} = [\begin{matrix} ​^{a} ω ​^{b} \\ ​^{a} v ​_{P}^{b} \end{matrix}] = [\begin{matrix} P (​^{a} V ​_{O}^{b}) \\ D (​^{a} V ​_{O}^{b}) + P (​^{a} V ​_{O}^{b}) \times r_{P ∕ O} \end{matrix}]

(3)

where r_P/O denotes the position of point P with respect to point O. Expression (3) implies that

​^{a} v ​_{O}^{b} = ​^{a} v ​_{O}^{b} + ​^{a} ω ​^{b} \times r_{P ∕ O} .

(4)

With reference to Fig. 2, the velocity state of the output platform with respect to the fixed one, six-dimensional vector ^AV^C_O=(^Aω^C,^Av^C_O) can be obtained through the middle and fixed platforms as

^{A} V_{O}^{C} =^{A} V_{O}^{B} +^{B} V_{O}^{C},

where ^AV^B_O=(^Aω^B,^Av^B_O) is the velocity state of the middle platform with respect to the fixed one, and ^BV^C_O=(^Bω^C,^Bv^C_O) is the velocity state between the output and middle platforms. Furthermore, these velocity states can be written in screw form as follows

​^{A} V ​_{O}^{B} = ​^{A} J ​_{i}^{B} + ​^{A} Ω ​_{i}^{B};

(6)

​^{B} V ​_{O}^{C} = ​^{B} J ​_{i}^{C} + ​^{B} Ω ​_{i}^{C};

(7)

and

​^{A} V ​_{O}^{C} = ​^{A} J ​_{i}^{C} + ​^{A} Ω ​_{i}^{C},

(8)

herein the local screw-coordinate Jacobian matrices are given by

\begin{array}{l} ^{A} J_{i}^{C} = [^{12} $_{i}^{13}^{13} $_{i}^{14}^{14} $_{i}^{15}^{15} $_{i}^{16}^{16} $_{i}^{17}^{17} $_{i}^{18}], \\ ^{B} J_{i}^{C} = [^{6} $_{i}^{7}^{7} $_{i}^{8}^{8} $_{i}^{9}^{9} $_{i}^{10}^{10} $_{i}^{11}^{11} $_{i}^{12}], \\ ^{A} J_{i}^{B} = [^{0} $_{i}^{1}^{1} $_{i}^{2}^{2} $_{i}^{3}^{3} $_{i}^{4}^{4} $_{i}^{5}^{5} $_{i}^{6}], \end{array}

where the matrices containing the joint-rate velocities are

\begin{array}{l} ^{A} Ω_{i}^{C} = [_{12} ω_{13}^{i}_{13} ω_{14}^{i}_{14} ω_{15}^{i}_{15} ω_{16}^{i}_{16} ω_{17}^{i}_{17} ω_{18}^{i}]^{T}, \\ ^{B} Ω_{i}^{C} = [_{6} ω_{7}^{i}_{7} ω_{8}^{i}_{8} ω_{9}^{i}_{9} ω_{10}^{i}_{10} ω_{11}^{i}_{11} ω_{12}^{i}]^{T}, \\ ^{A} Ω_{i}^{B} = [_{0} ω_{1}^{i}_{1} ω_{2}^{i}_{2} ω_{3}^{i}_{3} ω_{4}^{i}_{4} ω_{5}^{i}_{5} ω_{6}^{i}]^{T} . \end{array}

3.1.1 Forward velocity analysis

In order to calculate $^{A} V _{O}^{B}$ , consider that ${\dot{\bar{q}}}_{1} = ^{2} ω _{3}^{1}$ , ${\dot{\bar{q}}}_{2} = ^{2} ω _{3}^{2}$ and ${\dot{\bar{q}}}_{3} = ^{2} ω _{3}^{3}$ are generalized speeds of the manipulator. Later on, with the systematic application of the Klein form of each one of the screws $^{4} $_{i}^{5}$ and $^{3} $ _{i}^{4}$ to both sides of expression(6), with the corresponding reduction of terms, one obtains

{​^{4} $_{i}^{5}; ​^{A} V ​_{O}^{B}} = {\dot{\bar{q}}}_{i},

(9)

and

{​^{3} $_{i}^{4}; ​^{A} V ​_{O}^{B}} = 0.

(10)

Casting into a matrix-vector form expressions (9) and (10), the input/output equation of velocity of the middle platform with respect to the fixed results in

\bar{J} ​^{A} V_{O}^{B} = \bar{G} ​ \dot{\bar{q}},

(11)

therein J̄ is the active screw-coordinate Jacobian matrix of B with respect to A which is computed as

\bar{J} = {[\begin{array}{l} ​^{4} $ ​_{1}^{5} & ​^{4} $ ​_{2}^{5} & ​^{4} $ ​_{3}^{5} & ​^{3} $ ​_{1}^{4} & ​^{3} $ ​_{2}^{4} & ​^{3} $ ​_{3}^{4} \end{array}]}^{T} ​ Δ,

(12)

where $Δ = [\begin{matrix} O & I \\ I & O \end{matrix}]$ is an operator of polarity defined by the identity and zero matrices, $I$ and $O$ respectively, of dimension $3 \times 3$ . Furthermore, $\bar{G} = I_{6}$ is the first order influence coefficients matrix between platforms B and A, while $\dot{\bar{q}} = {[{\dot{\bar{q}}}_{1} {\dot{\bar{q}}}_{2} {\dot{\bar{q}}}_{3} 0 0 0]}^{T}$ is the first order driving matrix associated to the generalized coordinates ${\bar{q}}_{i}$ .

On the other hand, in order to find $^{A} V _{O}^{C}$ , consider that ${\dot{\underline{q}}}_{1} = ^{14} ω _{15}^{1}$ , ${\dot{\underline{q}}}_{2} = ^{14} ω _{15}^{2}$ and ${\dot{\underline{q}}}_{3} = ^{14} ω _{15}^{1}$ are generalized speeds of the robot. The systematic application of the Klein form of the screws $^{16} $ _{i}^{17}$ to both sides of expression (8) leads to

{16 $_{i}^{17}; ​^{A} V ​_{O}^{C}} = {\dot{\underline{q}}}_{i},

(13)

where the systematic application of the Klein form of the screws ⁹$₁¹⁰ to both sides of expression (5) yields

{​^{9} $ ​_{i}^{10}; ​^{A} V ​_{O}^{C}} = {​^{9} $ ​_{1}^{10}; ​^{A} V ​_{O}^{B}} .

(14)

After, casting into a matrix-vector form expressions (13) and (14),the input/output equation of velocity of the output platform C with respect to the fixed platform A results in

\underline{J} ​ ​^{A} V ​_{O}^{C} = \underline{G} ​ \underline{\dot{q}},

(15)

where

\underline{J} = {[​^{16} $ ​_{1}^{17} ​^{16} $ ​_{2}^{17} ​^{16} $ ​_{3}^{17} ​^{9} $ ​_{1}^{10} ​^{9} $ ​_{2}^{10} ​^{9} $ ​_{3}^{10}]}^{T} Δ

(16)

is the modified active screw-coordinate Jacobian matrix of C with respect to A. Furthermore, the first order influence coefficients matrix G between bodies C and A is given by

\underline{G} = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & {^{9} $_{1}^{10};^{A} V_{O}^{B}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {^{9} $_{2}^{10};^{A} V_{O}^{B}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {^{9} $_{3}^{10};^{A} V_{O}^{B}} \end{matrix}],

while $\underline{\dot{q}} = {[{\dot{\underline{q}}}_{1} {\dot{\underline{q}}}_{2} {\dot{\underline{q}}}_{3} 1 1 1]}^{T}$ is the first order driving matrix of the robot concerned with the generalized coordinates q _i .

3.2 Acceleration analysis

The reduced acceleration state, or accelerator for brevity, of a body b with respect to another body or reference frame a is defined as a six-dimensional vector ^aA^b_O given by

​^{a} A ​_{O}^{b} \equiv [\begin{matrix} ​^{a} \dot{ω} ​^{b} \\ ​^{a} a ​_{O}^{b} - ​^{a} ω ​_{O}^{b} \times ​^{a} v ​_{O}^{b} \end{matrix}],

(17)

where $^{a} {\dot{ω}}^{b}$ and ^aa^b_O are, respectively, the angular and linear accelerations considering point O as the reference pole. On the other hand, assuming that P is a point of interest of body b, then the accelerator ^a A ^b _p can be obtained applying the concept of helicoidal vector field as

​^{a} A ​_{P}^{b} = [\begin{matrix} P (​^{a} A ​_{O}^{b}) \\ D (​^{a} A ​_{O}^{b}) + P (​^{a} A ​_{O}^{b}) \times r_{P ∕ O} \end{matrix}]

(18)

Expression (18) implies that

​^{a} a ​_{P}^{b} - ​^{a} ω ​^{b} \times ​^{a} v ​_{P}^{b} = ​^{a} a ​_{O}^{b} -^{a} ω^{b} \times^{a} v_{O}^{b} + ​^{a} \dot{ω} ​^{b} \times r_{P ∕ O}

(19)

Therefore

​^{a} a ​_{P}^{b} = ​^{a} a ​_{O}^{b} + ​^{a} ω ​^{b} \times (​^{a} v ​_{P}^{b} - ​^{a} v ​_{O}^{b}) + ​^{a} \dot{ω} ​^{b} \times r_{P ∕ O}

(20)

With reference to Fig. 2, the accelerator of the output platform with respect to the fixed platform, six-dimensional vector ^AA^C_O= (^Aω^C, ^Aa^C_O- ^Aω^C × ^Av^C_O), can be obtained through the middle and fixed platforms as

​^{A} A ​_{O}^{C} = ​^{A} A ​_{O}^{B} + ​^{B} A ​_{O}^{C} + [\begin{matrix} ​^{A} V ​_{O}^{B} & ​^{B} V ​_{O}^{C} \end{matrix}],

(21)

where the brackets [* *] denote the Lie product of the Lie algebra e(3) of the Euclidean group E(3). Moreover, $^{A} A _{O}^{B} = (^{A} \dot{ω} ^{B}, ^{A} a _{O}^{B} - ^{A} ω ^{B} \times ^{A} v _{O}^{B})$ is the accelerator of the middle platform with respect to the fixed platform, while $^{B} A _{O}^{C} = (^{B} \dot{ω} ^{C}, ^{B} a _{O}^{C} - ^{B} ω ^{C} \times ^{B} v _{O}^{C})$ is the accelerator of the output platform with respect to the middle one. Furthermore, these accelerators can be written in screw form as follows

\begin{array}{l} ​^{A} A ​_{O}^{C} = ​^{A} J ​_{i}^{C} ​ ​^{A} \dot{Ω} ​_{i}^{C} ​ + ​^{A} L ​_{i}^{C} \\ ​^{A} A ​_{O}^{B} = ​^{A} J ​_{i}^{B} ​ ​^{A} \dot{Ω} ​_{i}^{B} ​ + ​^{A} L ​_{i}^{B}, \\ ​^{B} A ​_{O}^{C} = ​^{B} J ​_{i}^{C} ​ ​^{B} \dot{Ω} ​_{i}^{C} ​ + ​^{B} L ​_{i}^{C} \end{array}}

(22)

where $^{A} \dot{Ω} _{i}^{C}$ , $^{A} \dot{Ω} _{i}^{B}$ and $^{B} Ω _{i}^{C}$ are matrices containing the joint‐rate accelerations of the robot, whereas $^{A} L _{i}^{C}$ , $^{A} L _{i}^{B}$ and $^{B} L _{i}^{C}$ are composed Lie products that are computed as

\begin{array}{l} ​^{A} L ​_{i}^{C} = \sum_{j = 12}^{16} [​^{j} ω ​_{j + 1}^{i} ​ ​^{j} $ ​_{i}^{j + 1} \sum_{k = j + 1}^{17} ​^{k} ω ​_{k + 1}^{i ​} ​^{k} $ ​_{i}^{k + 1}] \\ ​^{B} L ​_{i}^{C} = \sum_{j = 6}^{10} [​^{j} ω ​_{j + 1}^{i} ​ ​^{j} $ ​_{i}^{j + 1} \sum_{k = j + 1}^{11} ​^{k} ω ​_{k + 1}^{i ​} ​^{k} $ ​_{i}^{k + 1}] . \\ ​^{A} L ​_{i}^{B} = \sum_{j = 0}^{4} [​^{j} ω ​_{j + 1}^{i} ​ ​^{j} $ ​_{i}^{j + 1} \sum_{k = j + 1}^{5} ​^{k} ω ​_{k + 1}^{i} ​^{k} $ ​_{i}^{k + 1}] \end{array}}

(23)

Following the trend of the velocity analysis, by applying reciprocal screw theory, the input/output equation of acceleration between the middle and fixed platforms results in

\bar{J} ​ ​^{A} A ​_{O}^{B} = ​^{A} H ​^{B} ​ \dot{\bar{q}} + ​^{A} C ​^{B},

(24)

where <^AH^B = I₆ is the higher order influence coefficients matrix between platforms B and A, while $\dot{\bar{q}} = {[{\dot{\bar{q}}}_{1} {\dot{\bar{q}}}_{2} {\dot{\bar{q}}}_{3} 0 0 0]}^{T}$ is the higher order driving matrix associated to the generalized coordinates q̄_i. Furthermore, ^AC^B is the complementary matrix of acceleration between bodies B and A which is given by

​^{A} C ​^{B} = [\begin{matrix} \begin{array}{l} {​^{4} $ ​_{1}^{5}; ​^{A} L ​_{​^{1}}^{B}} \\ {​^{4} $ ​_{2}^{5}; ​^{A} L ​_{2}^{C}} \\ {​^{4} $ ​_{3}^{5}; ​^{A} L ​_{3}^{C}} \end{array} \\ \begin{array}{l} {​^{3} $ ​_{1}^{4}; ​^{A} L ​_{​^{1}}^{B}} \\ {​^{3} $ ​_{2}^{4}; ​^{A} L ​_{2}^{C}} \\ {​^{3} $ ​_{3}^{4}; ​^{A} L ​_{3}^{C}} \end{array} \end{matrix}]

(25)

On the other hand, the input/output equation of acceleration of the output platform with respect to the fixed platform is obtained as

\underline{J} ​ ​^{A} A ​_{O}^{C} = A H^{C} ​ \underline{\dot{q}} + ​^{A} C ​^{C},

(26)

where ^AH^C is the higher order influence coefficients matrix between platforms B and C which is given by

​^{A} H ​^{C} = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & {​^{9} $ ​_{1}^{10}; ​^{A} A ​_{O}^{B}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {​^{9} $ ​_{2}^{10}; ​^{A} A ​_{O}^{B}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {​^{9} $ ​_{3}^{10}; ​^{A} A ​_{O}^{B}} \end{matrix}]

while ${\underline{q}}_{i}$ is the higher order driving matrix concerned with the generalized coordinates q _i . Furthermore, ^AC^C is the complementary matrix of acceleration between bodies C and A which is computed as

​^{A} C ​^{C} = [\begin{array}{l} {​^{16} $ ​_{1}^{17}; ​^{A} L ​_{​^{1}}^{C}} \\ {​^{16} $ ​_{2}^{17}; ​^{A} L ​_{2}^{C}} \\ {​^{16} $ ​_{3}^{17}; ​^{A} L ​_{3}^{C}} \\ {​^{9} $ ​_{1}^{10}; [​^{A} V ​_{O}^{B} ​ ​ ​^{B} V ​^{C}] + ​^{B} L ​_{​^{1}}^{C}} \\ {​^{9} $ ​_{2}^{10}; [​^{A} V ​_{O}^{B} ​ ​ ​^{B} V ​^{C}] + ​^{B} L ​_{​^{2}}^{C}} \\ {​^{9} $ ​_{3}^{10}; [​^{A} V ​_{O}^{B} ​ ​ ​^{B} V ​^{C}] + ​^{B} L ​_{​^{3}}^{C}} \end{array}]

(27)

4. Dynamic analysis

In this section, a harmonious combination of the theory of screws and the principle of virtual work allows systematically obtainingthe kinetostatic equations of the robot under study.

Let cm be the centre of mass of body b. A general wrench of body b, as it is observed from another body or reference frame a, is defined as a six-dimensional vector ^aW^b_cm given by

​^{a} W ​_{c m}^{b} = [\begin{matrix} ​^{a} f ​^{b} \\ ​^{a} n ​_{c m}^{b} \end{matrix}],

(28)

where ^af^b is a pure force applied to body b, whereas ^aη^b_cm is the moment produced by f about point cm. Assuming that g is the acceleration of gravity, then it immediately emerges that the wrench associated to the mass m of body b affected by vector g , ^aW̄^b_cm, is given by

​^{a} \bar{W} ​_{c m}^{b} = [\begin{matrix} m g \\ 0 \end{matrix}] ​ .

(29)

Moreover, according to D'Alembert's principle, the inertial wrench between bodies b and a is computed as

​^{a} \overset{⌣}{W} ​_{c m}^{b} = [\begin{matrix} - m ​ ​^{a} a ​_{c m}^{b} \\ - ​^{a} I ​_{c m}^{b} ​ ​^{a} \dot{ω} ​^{b} - ​^{a} ω ​^{b} \times ​^{a} I ​_{c m}^{b} ​^{a} ω ​^{b} \end{matrix}]

(30)

where ^aI^b_cm is the centroidal inertia matrix regarding to the fixed reference frame. Matrix ^aI^b_cm can be calculated by transforming the inertia matrix expressed in a reference frame embedded in body 6, I^b_cm, through the rotation matrix ^aR^b between bodies b and a as

​^{a} I ​_{c m}^{b} = ​^{a} R ​^{b} I_{c m}^{b} {[​^{a} R ​^{b}]}^{T}

(31)

Furthermore, let ^af^b_cm and ^an^b_cm be a pure force and a moment applied to the mass of centre of body b. Then, it is necessary to consider an external wrench given by

​^{a} W ​_{c m}^{b} = [\begin{matrix} ​^{a} f ​_{c m}^{b} \\ ​^{a} n ​_{c m}^{b} \end{matrix}] .

(32)

Then, the total wrench results in

​^{a} W ​_{c m}^{b} = ​^{a} \bar{W} ​_{c m}^{b} + ​^{a} \overset{⌣}{W} ​_{c m}^{b} + ​^{a} W ​_{c m}^{b} .

(33)

On the other hand, according to Gallardo et al. [17], the power ^aw^b_cm produced by the resulting wrench ^aW^b_cm and the velocity state ^aV^b_cm, which has its centre of mass as pole of reference, can be determined using the Klein form as follows

​^{a} w ​_{c m}^{b} = {​^{a} W ​_{c m}^{b}; ​^{a} V ​_{c m}^{b}} .

(34)

In order to apply the principle of virtual work, it is necessary to express the velocity state ^aV^b_cm in terms of the generalized speeds ${\dot{\bar{q}}}_{i}$ and ${\dot{\underline{q}}}_{i}$ . of the robot. In that way, it is straightforward to demonstrate that the velocity state of the middle platform with respect to the fixed, taking the centre of mass of body B as the reference pole, ^AV^B_cm, can be written according to expressions (3) and (11) as

​^{A} V ​_{c m}^{B} = [\begin{matrix} ​^{A} Ρ ​^{B} & \dot{\bar{q}} \\ ​^{A} ​ D^{B} & \dot{\bar{q}} \end{matrix}]

(35)

where ^AP^B = [^AP^B_ij] and ^AD^B = [^Ad^B_ij] are, respectively, the 3 × 6 primal influence coefficients matrix and the 3 × 6dual influence coefficients matrix of body B with respect to body A. Then, considering ^AW^B_cm =(^Af^B,^An^B_cm), where ^Af^B= (^Af^B_X,^Af^B_Y,^Af^B_Z) and ^An^B_cm = (^An^B_X,^An^B_Y,^An^B_Z), as the total wrench of body B with respect to body A, the power of A by applying Eq. (34) collecting terms results in

​^{A} w ​_{c m}^{B} = {\bar{λ}}_{1}^{B} ​ {\dot{\bar{q}}}_{1} + {\bar{λ}}_{2}^{B} ​ {\dot{\bar{q}}}_{2} + {\bar{λ}}_{3}^{B} ​ {\dot{\bar{q}}}_{3},

(36)

where the multipliers ${\bar{λ}}_{i}^{B}$ are given by

\begin{array}{l} {\bar{λ}}_{i}^{B} = ​^{A} d ​_{1 i}^{B} ​ ​^{A} f ​_{X}^{B} + ​^{A} d ​_{2 i}^{B} ​ ​^{A} f ​_{Y}^{B} + ​^{A} d ​_{3 i}^{B} ​ ​^{A} f ​_{Z}^{B} + \\ + ​^{A} p ​_{1 i}^{B} ​ ​ ​^{A} n ​_{X}^{B} + ​^{A} p ​_{2 i}^{B} ​ ​^{A} n ​_{Y}^{B} + ​^{A} p ​_{3 i}^{B} ​ ​^{A} n ​_{Z}^{B} \end{array}

(37)

Similarly, the power ^Aw^c_cm of the platform C is computed as

​^{A} w_{c m}^{C} = {\underline{λ}}_{1}^{C} ​ {\dot{\underline{q}}}_{1} + {\underline{λ}}_{2}^{C} ​ {\dot{\underline{q}}}_{2} + {\underline{λ}}_{3}^{C} ​ {\dot{\underline{q}}}_{3} .

(38)

Furthermore, it is straightforward to demonstrate that the power ^Aw^b_cm of any link b of the robot can be expressed as

​^{A} w_{c m}^{b} = {\bar{λ}}_{1}^{b} ​ {\dot{\bar{q}}}_{1} + {\bar{λ}}_{2}^{b} ​ {\dot{\bar{q}}}_{2} + {\bar{λ}}_{3}^{b} ​ {\dot{\bar{q}}}_{3} + {\underline{λ}}_{1}^{b} ​ {\dot{\underline{q}}}_{1} + {\underline{λ}}_{2}^{b} ​ {\dot{\underline{q}}}_{2} + {\underline{λ}}_{3}^{b} ​ {\dot{\underline{q}}}_{3} .

(39)

After, the total work W of the robot is given by

\begin{array}{l} W = ({\bar{f}}_{1} + {\bar{λ}}_{1}^{B} + \sum^{b} {\bar{λ}}_{1}^{b}) {\dot{\bar{q}}}_{1} + ({\bar{f}}_{2} + {\bar{λ}}_{2}^{B} + \sum^{b} {\bar{λ}}_{2}^{b}) {\dot{\bar{q}}}_{2} + \\ + ​ ({\bar{f}}_{3} + {\bar{λ}}_{3}^{B} + \sum^{b} {\bar{λ}}_{3}^{b}) {\dot{\bar{q}}}_{3} + ({\underline{f}}_{1} + {\underline{λ}}_{1}^{C} + \sum^{b} {\underline{λ}}_{1}^{b}) {\dot{\underline{q}}}_{1} + \\ + ​ ({\underline{f}}_{2} + {\underline{λ}}_{2}^{C} + \sum^{b} {\underline{λ}}_{2}^{b}) {\dot{\underline{q}}}_{2} + ({\underline{f}}_{3} + {\underline{λ}}_{3}^{C} + \sum^{b} {\underline{λ}}_{3}^{b}) {\dot{\underline{q}}}_{3} \end{array}

(40)

where ${\bar{f}}_{i}$ and ${\underline{f}}_{i}$ are the forces of the active prismatic joints.

The principle of virtual work states that if a multi-body system is in equilibrium under the effect of external actions, then the global work produced by the external forces with any virtual velocity must be zero, for details the reader is referred to [11]. Introducing the generalized virtual velocities $δ {\dot{\bar{q}}}_{i}$ , and $δ {\dot{\underline{q}}}_{i}$ , and taking into account that the work due to the internal reactions of constraint vanishes, the total virtual work δW, is given by

\begin{array}{l} δ W = ({\bar{f}}_{1} + {\bar{λ}}_{1}^{B} + \sum^{b} {\bar{λ}}_{1}^{b}) δ {\dot{\bar{q}}}_{1} + ({\bar{f}}_{2} + {\bar{λ}}_{2}^{B} + \sum^{b} {\bar{λ}}_{2}^{b}) δ {\dot{\bar{q}}}_{2} + \\ ({\bar{f}}_{3} + {\bar{λ}}_{3}^{B} + \sum^{b} {\bar{λ}}_{3}^{b}) δ {\dot{\bar{q}}}_{3} + ({\underline{f}}_{1} + {\underline{λ}}_{1}^{C} + \sum^{b} {\underline{λ}}_{1}^{b}) δ {\dot{\underline{q}}}_{1} + \\ ({\underline{f}}_{2} + {\underline{λ}}_{2}^{C} + \sum^{b} {\underline{λ}}_{2}^{b}) δ {\dot{\underline{q}}}_{2} + ({\underline{f}}_{3} + {\underline{λ}}_{3}^{C} + \sum^{b} {\underline{λ}}_{3}^{b}) δ {\dot{\underline{q}}}_{3} = 0 ​ . \end{array}

(41)

Finally, taking into account that the generalized virtual velocities are arbitrary then, in order to satisfy Eq. (41), the generalized forces are obtained as

{\bar{f}}_{i} = - {\bar{λ}}_{i}^{B} - \sum^{b} {\bar{λ}}_{i}^{b},

(42)

and

{\underline{f}}_{i} = - {\underline{λ}}_{i}^{C} - \sum^{b} {\underline{λ}}_{i}^{b} .

(43)

5. Numerical example

In this section a numerical example dealing with the forward kinematics and dynamics of the manipulator is presented as a case study. Using hereafter SI units, the parameters of the robot are sketched in Table 1.

Table 1.

Parameters of the numerical example

û₁ = û'₁	k̂
û₂ = û'₂	0.866î −0.5k̂
û₃ = û'₃	−0.866î −0.5k̂
U ₁	(0.508,0,0)
U ₂	(−0.254, 0, −0.440)
U ₃	(−0.254, 0, 0.440)
R ₁	(0.152,0,0)
R ₂	(−0.076,0,-0.132)
R ₃	(−0.076,0,0.132)

Furthermore, the coordinates of the spherical and revolute joints of the robot at its home position, also known as its reference configuration, are provided in Table 2.

Table 2.

Home position of the robot

s ₁	(0.152,-0.635,0)
s ₂	(−0.076,-0.635,-0.132)
s ₃	(−0.076,-0.635,0.132)
r ₁	(0.076, −0.635,0)
r ₂	(−0.038, −0.635, −0.066)
r ₃	(−0.038, −0.635,0.066)
S ₁	(0.254,-1.270,0)
s ₂	(−0.127,-1.270,-0.220)
S₃	(−0.127,-1.270,0.220)
S' ₁	(0.076,-1.270,0)
s' ₂	(0.038, −1.270, −0.066)
s' ₃	(0.038, −1.270,0.066)

On the other hand, the inertial and mass properties of the active RPS-type and UPS-type limbs are listed, respectively, in Tables 3 and 4.

Table 3.

Inertial parameters of the active RPS-type limbs

Body	Label	Mass	I_u	I_mm	I_nn
Middle platform	6	7.028	0.0433	0.0862	0.0433
Rod	3	1.681	0.0172	1.65E-04	0.0172
Cylinder	2	2.758	0.0182	5.86E-04	0.0182

Table 4.

Inertial parameters of the UPS-type limbs

Body	Label	Mass	I_LL	I_MM	I_NN
Output platform	18	24.403	0.338	0.675	0.338
Rod	15	3.86	0.158	4.34E-04	0.158
Cylinder	14	9.803	0.3.11	2.73E-03	0.311

It must be noted that the data provided in Tables 3 and 4 are expressed in local reference frames attached to the corresponding moving bodies. The mass and inertial properties of the passive kinematic chains are neglected.

Assuming that the generalized coordinates are constrained to the following periodical functions

\begin{array}{l} {\bar{q}}_{1} = 0.635 - 0.101 \sin (t) \\ {\bar{q}}_{2} = 0.635 - 0.114 \sin (t) \\ {\bar{q}}_{3} = 0.635 - 0.114 \sin (t) \cos (t) \\ {\underline{q}}_{1} = 1.295 + 0.076 \sin (t) \cos (0.25 t^{2}) \\ {\underline{q}}_{2} = 1.295 - 0.127 \sin^{2} (t) \\ {\underline{q}}_{3} = 1.295 + 0.127 \sin^{3} (t) \sin (0.1 t^{2}) \end{array}},

where the interval for the time t is chosen as 0 ≤ t ≤ 2π. The first part of the example consists of finding the angular and linear velocities of the centre of the output platform with respect to the fixed platform. The corresponding plots of the velocity analysis, using screw theory, are shown in Figs. 4 and 6.

Figure 4.

Translational velocity, calculated with screw theory, of the centre of the output platform

Figure 5.

Translational velocity, calculated using ADAMS©, of the centre of the output platform

Figure 6.

Angular velocity, calculated with screw theory, of the output platform

The next part of the exercise consists of finding the angular and linear accelerations of the centre of the output platform with respect to the fixed frame. These plots using screw theory are provided in Figs. 8 and 10. The last part of the numerical example demands the computation of the generalized forces q̄_i and q _i . These results are given in Figs. 12 and 14.

Finally, the results of the numerical example are compared in Figs. 5, 7, 9, 11, 13 and 15, with simulations generated with the aid of specialized software like ADAMS©.

Figure 7.

Angular velocity, calculated using ADAMS©, the output platform

Figure 8.

Translational acceleration, calculated with screw theory, of the centre of the output platform

Figure 9.

Linear acceleration, calculated using ADAMS©, of the centre of the output platform

Figure 10.

Angular acceleration, calculated with screw theory, of the output platform

Figure 11.

Angular acceleration, calculated using ADAMS©, of the output platform

Figure 12.

Generalized forces of the inner active manipulator 3RPS, using screw theory and the principle of virtual work

Figure 13.

Generalized forces of the inner active manipulator 3RPS, using ADAMS©

Figure 14.

Generalized forces of the active manipulator 3UPS, using screw theory and the principle of virtual work

Figure 15.

Generalized forces of the active manipulator 3UPS, using ADAMS©

6. Conclusions

In this contribution the dynamic analysis, including speed and acceleration, of a series-parallel manipulator has been solved using screw theory and the principle of virtual work.

The velocity and acceleration states of the moving platforms, with respect to the fixed frame, are written in screw form through each one of the active limbs of the manipulator. The expressions obtained are linear, simple and compact through the use of the Klein form of Lie algebra.

A significant advantage of this formulation is that it is not necessary to know the passive velocities and accelerations to solve forward velocity and acceleration analysis, respectively.

Finally, the driver forces required to obtain the desired velocity and acceleration states for the output platform are calculated considering both the theory of screws and the principle of virtual work. The resulting equations are simple and linear, unlike those obtained by the Newton-Euler method, the Lagrangian and some others.

With the proposed method, it is not necessary to determine the instantaneous values of the internal reactions of the mechanism or to calculate the energy of the entire system. With these results it is possible to size motors/actuators of the manipulator for a given application.

Footnotes

7. Acknowledgments

This work was supported by Consejo Nacional de Ciencia y Tecnología of Mexico.

References

Geng

Haynes

Lee

Carroll

On the dynamic model and kinematics analysis of a class of Stewart platforms. Robot. Auton. Syst. 9, 237–254 (1992)

Liu

Lewis

Lebret

Taylor

The singularities and dynamics of a Stewart platform manipulator. J. Intell. Robot. Syst. 8, 287–308 (1993)

Dasgupta

Mruthyunjaya

A Newton–Euler formulation for the inverse dynamics of the Stewart platform manipulator. Mech. Mach. Theory 34, 711–725 (1998)

Dasgupta

Choudhury

A general strategy based on the Newton–Euler approach for the dynamic formulation of parallel manipulators. Mech. Mach. Theory 34, 801–824 (1999)

Carvalho

Ceccarelli

A closed-form formulation for the inverse dynamics of a Cassino parallel manipulator. Multibody Syst. Dyn. 5, 185–210 (2001)

Di. Gregorio

Parenti-Castelli

Dynamics of a class of parallel wrists. ASME J. Mech. Des. 126, 436–441 (2004)

Rao

A.B.K.

Saha

S.K.

Rao

P.V.M.

Dynamics modelling of hexaslides using the decoupled natural orthogonal complement matrices. Multibody Syst. Dyn. 15, 159–180 (2006)

Lopes

A.M.

Almeida

The generalized momentum approach to the dynamic modeling of a 6-dof parallel manipulator. Multibody Syst. Dyn. 21, 123–146 (2009)

Zhang

C.-D.

Song

S.-M.

An efficient method for the inverse dynamics of manipulators based on the virtual work principle. J. Robotic Syst. 10 605–627 (1993)

10.

Wang

Gosselin

A new approach for the dynamic analysis of parallel manipulators. Multibody Syst. Dyn. 2 317–334 (1998)

11.

Tsai

L.W.

Robot Analysis. The Mechanics of Serial and Parallel Manipulators, John Wiley and Sons Inc, New York (1999)

12.

Tsai

L.-W.

Solving the inverse dynamics of a Stewart-Gough manipulator by the principle of virtual work. ASME J. Mech. Des. 122, 3–9 (2000)

13.

Staicu

Zhang

A novel dynamic modelling approach for parallel mechanisms analysis. Robot. CIM-Int. Manuf 24, 167–172 (2008)

14.

Staicu

Dynamics of the spherical 3-UPS/S parallel mechanism with prismatic actuators. Multibody Syst. Dyn. 22, 115–132 (2009)

15.

Zhao

Gao

Inverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual work. Robotica 27, 259–268 (2009)

16.

Akbarzadeh

Enferadi

A virtual work based algorithm for solving direct dynamics problem of a 3-RRP spherical parallel manipulator. J. Intell. Robot. Syst. Published online 2010.

17.

Gallardo

Rico

J.M.

Frisoli

Checcacci

Bergamasco

Dynamics of parallel manipulators by means of screw theory. Mech. Mach. Theory 38, 1113–1131 (2003)

18.

Gallardo-Alvarado

Aguilar-Nájera

C.R.

Casique-Rosas

Pérez-González

Rico-Martnez

J.M.

Solving the kinematics and dynamics of a modular spatial hyper-redundant manipulator by means of screw theory, Multibody Syst. Dyn. 20, 307–325, (2008)

19.

Gallardo-Alvarado

Orozco-Mendoza

Rico-Martinez

J.M.

A novel five-degrees-of-freedom decoupled robot. Robotica 28, 909–917 (2010)

20.

Gough

V.E.

Contribution to discussion to papers on research in automobile stability and control and in type performance. Proceedings Automation Division Institution of Mechanical Engineers 1, 392–5(1957)

21.

Gough

V.E.

Whitehall

S.G.

Universal tyre test machine. Proceedings of the FISITA Ninth International Technical Congress. (1962) pp.117–37.

22.

Stewart

D. A.

Platform with six degrees of freedom. Proc. Inst. Mech. Engrs. I, 371–386 (1965)

23.

Innocenti

Forward kinematics in polynomial form of the general Stewart platform. ASME J. Mech. Des. 123, 254–260 (2001).

24.

Gallardo

Orozco

Rodrguez

Finding the jerk properties of multibody systems using helicoidal vector fields. Proc. Inst. Mech. Engrs, IMechE Part C J. Mech. Eng. Sci., 222, 2217–2229 (2008)