Geometry and kinematics for a spherical-base integrated parallel mechanism

Figure 1 illustrates a three-fingered metamorphic robotic hand grasping and manipulating a disk. The metamorphic robotic hand (Metahand) contains a reconfigurable palm and three two-phalanx fingers. The reconfigurable palm is formed by a spherical five-bar linkage, with link AE as a grounded link, and the other four links are symmetrically arranged with respect to link AE such that links AB and ED are of the same length and so do links BC and DC. The three fingers are respectively mounted on link AE at point A₁, on link DC at point A₂ and on link BC at point A₃. When the palm is in a configuration that all the links are in a same plane, the three points A₁, A₂ and A₃ are evenly distributed about point O, i.e. centre of the spherical five-bar linkage. The 2-DOF reconfigurable palm changes configuration of the whole hand, and increases workspace, dexterity and manipulability of the hand [30, 31]. The idea of equivalence about considering the multi-fingered hand with a grasped object as a parallel mechanism was put forward by Borras-Sol and Dollar [33, 34]. When the hand is used to grasp and manipulate an object such as a disk as shown in Fig. 1, if the contact points between the object and the fingertips are thought of as spherical joints, an equivalent parallel mechanism with a reconfigurable base can be intuitively generated as illustrated in Fig. 2. This parallel mechanism is coined in this paper spherical-base integrated parallel mechanism.

As shown in Fig. 2, the spherical-base integrated parallel mechanism is composed of a spherical reconfigurable base, a moving platform and three identical revolute–revolute-spherical chain connecting them. The reconfigurable base consists of five links which connect to each other forming a spherical five-bar linkage. In this design, link AE is fixed and joints A and E are assumed to be active joints to change the configurations of the base, and joints B, C and D are passive joints. The axes of these five joints always intersect at point O. The angles covered by links AB, BC, CD, DE and EA are denoted as φ ₁ throughout φ ₅ separately, and the sum of which satisfies φ ₁ + φ ₂ + φ ₃ + φ ₄ + φ ₅ = 2π. There are three identical limbs mounted at point A _i (i = 1, 2 and 3), and the angles between OA₁ and OA, OB and OA₂, OA₃ and OD are indicated as δ ₁, δ ₂ and δ ₃. The angle between any two limbs is 120° in the initial configuration of the mechanism when the five links of the reconfigurable base are located in a same horizontal plane. However, one has to clarify that this initial state of the mechanism as a singular configuration is suitable for theoretical computations rather than a starting configuration for practical applications. Each limb is made up of two linkages coupled by a revolute joint B _i (i = 1, 2 and 3). The limbs are connected to the reconfigurable base by revolute joints A _i (i = 1, 2 and 3) and the moving platform by spherical joints C _i (i = 1, 2 and 3). The length of link A _i B _i is denoted as l _i1, while that of link B _i C _i is denoted as l _i2 (i = 1, 2 and 3).

As shown in Fig. 2, a global coordinate system F {O-xyz} is attached to the reconfigurable base with point O as the origin and its y-axis directed towards the upper platform and perpendicular to the plane formed by the axes of joints A and E. The z-axis of the coordinate system lies along OA₁. The radius of the spherical base is set at 1 for simplifying the calculation. For solving the geometric relationship of the reconfigurable base, local coordinate systems M _i {O-x _i y _i z _i (i = 1 to 5)} are created at point O with the z _i -axis aligned the joint axis (joint A, B, C, D and E respectively) and the y _i -axis perpendicular to z _i  × z _i+1 (if i = 5, z _i+1-axis represents z ₁-axis). The coordinate system M₅ {O-x ₅ y ₅ z ₅} is taken as an example indicated in Fig. 3. In this local coordinate system, the position vectors of point A, B, C, D and E can be calculated as

where s and c denote the sine and cosine functions, θ ₁ to θ ₅ are the rotation angles of joints A to E, and \(\varvec{P}_{\text{E}} = (0,\,0,\,1)^{\text{T}}\).

Due to the geometric constraints of the spherical base, the position vectors of its joints have to satisfy the following equations,

Substituting Eqs. (2), (3) and (4) into Eqs. (5) and (6) leads to the coordinates of x _C and y _C represented in terms of z _C as where

Substituting Eqs. (8) and (9) into Eq. (7) results in a quadratic equation aswhere S ₁ = Q ² + N ² + 1, S ₂ = 2 (PQ + MN) and S ₃ = P ² + M ² − 1.

Solving Eq. (10), the coordinate of z _C can be obtained as

Thus, the value of joint angle θ ₄ is obtained by substituting Eq. (11) into z _C of Eq. (3) as

The two possible values of z _C result in two joint angle θ ₄, leading to two configurations of ∆BCD, one of which represents the case that the triangle vertex C appears above BD and the other indicates the case when vertex C is below BD.

Apart from Eq. (3), the position vector of point C can be expressed in another form aswhich leads to another expression of z _C as

Substituting Eq. (14) into Eq. (11) and rearranging the equation yieldswhere,

Solving Eq. (15) gives the joint angle θ ₂ as

The above equation indicates two solutions for θ ₂, one of which implies the triangle vertex B locates below AC, and the other represents vertex B above AC. Because the reconfigurable base is a closed chain, the joint value θ ₁ and θ ₅ are not totally independent. When assigning the value of angle θ ₅, the spherical five-bar linkage mechanism degenerates to a spherical four-bar linkage mechanism. At that time, rotating joint A will make the spherical four-bar linkage mechanism reach its limited positions when point B, C and D lie in the same plane, thus it has,

The mechanism has two limited position as the link AD can rotate on both side with respect to link DC. Thus, the value θ ₅ decides the range of that of θ ₁, the relation between the two angles can be obtained as

Substituting Eqs. (1), (2) and (3) into Eqs. (19) throughout (21) and solving the latter gives the two limited values of angle θ ₁ aswhere \(U_{1} = - s\varphi_{4} c\theta_{5} c\varphi_{5} \text{s} \varphi_{1} - s\varphi_{1} c\varphi_{4} s\varphi_{5} ,\,U_{2} = s\varphi_{1} s\varphi_{4} s\theta_{5} \;{\text{and}}\;U_{3} = c\left( {\varphi_{2} + \varphi_{3} } \right) - c\varphi_{1} c\varphi_{4} c\theta_{5} + s\varphi_{4} c\varphi_{1} c\theta_{5} s\varphi_{5} .\)

Hence the range of θ ₁ with respect to θ ₅ iswhere

Based on the above analysis, it can be found that given a pair of θ ₁ and θ ₅, there are four groups of solution for θ ₂, θ ₃ and θ ₄ resulting in four different configurations of the base. Motion planning is needed when controlling this mechanism because the configuration of spherical base is considered by the order of its inputs.

A local coordinate system M′{O′-x′y′z′} is attached to the upper moving platform with the origin O′ coincided with the centroid of the equilateral triangle \(\Delta {\text{C}}_{ 1} {\text{C}}_{ 2} {\text{C}}_{ 3}\) and the z′-axis directed to point C₁. The coordinates of A _i (i = 1, 2 and 3) in the global coordinate system are given by,

The coordinates of C _i in the coordinate system M′ can be obtained as

The upper moving platform C₁C₂C₃ is an equilateral triangle as |O′C ₁| = |O′C ₂| = |O′C ₃| = 3r ². The position vector \({}^{\text{F}}\varvec{P}_{{{\text{C}}i}}\) of C _i (i = 1, 2 and 3) with respect to global coordinate frame F is given by the transformation as follows,where \({}^{\text{F}}\varvec{P}_{\text{OO'}}\) is the position vector of O′ expressed in the global coordinate frame F and \({}^{\text{F}}{\mathbf{R}}_{{{\text{M}}^{{\prime }} }}\) is the rotation matrix indicating the rotation of coordinates from coordinate frame M′ to the global coordinate frame F.

The sequence of calculating the forward kinematics of the spherical-base integrated mechanism is to take the configuration of the base into consideration primarily as a way to degenerate the whole mechanism into a 3-RRS mechanism with a confirmed base configuration, then apply the way to formulating forward kinematics of a parallel mechanism to this simplified parallel mechanism, which is well presented in the Ref. [35‐37]. For each limb in this proposed mechanism, the local limb coordinate system K_i {O-x _Ki y _Ki z _Ki } (i = 1, 2 and 3) is established with the origin O, z _Ki -axis directed to point A _i and y _Ki -axis perpendicular to the plane formed by the linkage of the reconfigurable base. In terms of Fig. 4, the y _K2 -axis is perpendicular to the plane constructed by ∆COD.

The position vector of point C _i in the global coordinate frame can be described aswhere k _i is the position vector of point C _i expressed in the local coordinate system M₅ {O-x ₅ y ₅ z ₅}, R _{K i} describes the transformation from the local limb coordinate system to the coordinate system M₅ {O-x ₅ y ₅ z ₅} as

The values of θ ₂ and θ ₄ can be obtained through Eqs. (16) and (12) according to the geometry constraints of the reconfigurable base. So k _i is computed by substituting θ ₂ and θ ₄ together with Eq. (32) into Eq. (31) as

For calculating the forward kinematics, the angle value of joint B _i is given, hence the length between point A _i and C _i is introduced by the following equation,

Thus, the upper moving platform, which is connected to the limbs by spatial joints, restricts the position of point C _i ; that is

Equation (35) describes the geometrical relation between the endpoints of any two limbs. So if i = 3, then i + 1 is equal to 1. By utilizing the way to calculating forward kinematics of the parallel mechanism introduced by Merlet [36], we can see that there are up to 16 solutions for the fixed-base parallel mechanism. However, in this spherical-base integrated mechanism, the number doubles as the reconfigurable base provides two configurations for given inputs investigated in Sect. 3.1. Path planning is necessary to get a desired configuration of the base and the moving platform.

The workspace of the spherical-base integrated parallel mechanism proposed in this paper is larger than that of the 3-RRS parallel mechanism with same limb and platform constructions and parameters. To make the comparison fairly, the base of the 3-RRS parallel mechanism is designed as the same of the initial configuration of the proposed mechanism (stated in Sect. 2.2) Under this definition, the workspace of this proposed mechanism is enlarged compared with the 3-RRS fixed-base parallel mechanism. By locking the base of this proposed mechanism in its initial state, it shares the same workspace of the 3-RRS parallel mechanism. When the joints of the base are released, it will contribute to a larger workspace as you can always lock the base during its motion in where the mechanism degenerates to a 3-RRS mechanism as a consequence. In other words, the workspace of the spherical-base integrated parallel mechanism is the sum of the workspaces of its degenerated 3-RRS parallel mechanisms with all possible base configurations.

The velocity of point C can be expressed as a linear combination of angular velocity of axis OA and OB or the other linear combination of angular velocity of axis OE and OD,

Since v _C is a passive variable, it should be estimated from Eqs. (52) and (53). Thus we take right inner product on both side of Eqs. (52) and (53) with P _D, it has

Substituting Eq. (55) into Eq. (54) yields,

Similarly, the angular velocity \(\dot{\theta }_{4}\) can be obtained and expressed as,

The Eqs. (56) and (57) can be expressed in matrix form as,

Thus the angular velocity of passive joints B and D is calculated through that of active joints A and E based on the geometric constraints of the spherical mechanism.

The screw algebra is introduced in this section for analysing the velocity of the spherical-base integrated parallel mechanism. A screw S is a six-dimensional vector representing instantaneous kinematic properties of a rigid body, commonly expressed as,

The first three components consist of a unit vector s directing along the screw axis, as well as the joint axis when describing a rotation. The last elements constitute s ₀ representing the moment of the vector s about the origin of the reference frame, and h expresses the screw pitch, which is equal to 0 for revolute joints and ∞ for prismatic joints, r is the position vector from the origin of the reference coordinate system to an arbitrary point on the screw axis s.

The whole mechanism Jacobian can be derived from the twist of the mechanism based on screw system notation. Figure 5a illustrates motion screws of the spherical-base integrated parallel mechanism. We can assume each limb as an open-loop chain connecting the end-effector to the base, as shown in Fig. 5b. Defining S _p as the instantaneous motion of the moving platform, the twist S _p can be derived from the linear combination of each joint’s twist within the loop. Referring to Fig. 5a, twist S _p can be obtained in terms of limb 1, 2 and 3 separately as

Substituting Eqs. (56) and (57) into Eqs. (61) and (62) respectively leads to, where According to [34], we know that the revolute-spherical screws dyad locate in a four-dimensional vector space. So the reciprocal screws form a two-system with zero pitch. Let \(\varvec{S}_{i1}^{r} \;{\text{and}}\;\varvec{S}_{i2}^{r} \quad \left( {i = \, 1,\,2\;{\text{and}}\;3} \right)\) denote a two-system of screws that is reciprocal to the four-system of screws S _i2 to S _i5 (i = 1, 2 and 3) of the ith limb. Performing the reciprocal product of both sides of Eqs. (60), (63) and (64) with reciprocal screw \(\varvec{S}_{i1}^{r} \;{\text{and}}\;\varvec{S}_{i2}^{r}\) gives six linear equations, which can be expressed in matrix form as,where and Δ denotes the reciprocal operator expressed as \(\Delta = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{I}} \\ {\mathbf{I}} & {\mathbf{0}} \\ \end{array} } \right].\)

The term \({\mathbf{J}}_{\theta i}\) is derived in detail in the Appendix 2, and \({\mathbf{J}}_{q}^{\text{T}}\), in general, is a 6 × 6 nonsingular matrix. Thus, multiplying both sides of Eq. (65) by the inverse of \({\mathbf{J}}_{q}^{\text{T}}\) gives the twist of the moving platform as,where ΔS _p is the twist of the moving platform with interchange of its primary part and secondary part comparing to S _p . The left-hand side and right-hand side of Eq. (65) gives the power of the platform and the actuated joints respectively, which provides background for dynamic analysis of the proposed parallel mechanism based on the concept of kinetic energy.

The spherical-base integrated parallel mechanism can be decomposed as three closed-loop mechanisms between any two of three limbs and an additional closed-loop of the five-bar reconfigurable base. The instantaneous motion of each link can be described as its instantaneous twist, and all the links’ motion in a closed-loop form a linear combination of all the instantaneous twists within the loop. Let twist S _ij denote the instantaneous motion along the jth joint in the ith limb and twist S _a, S _b, S _c, S _d and S _e denote that along the joints of the reconfigurable base. Define variable \(\dot{\theta }_{ij} \;{\text{and}}\;\dot{\theta }_{k}\) as the velocity of the jth joint in the ith limb and the velocity of kth joint in the reconfigurable base. Based on the notations introduced above, twists of three closed-loop-mechanisms are expressed as separately as follows.

For the closed-loop composed of limb 1 and 2, shown in Fig. 6a, the closed-loop-twist is

For limb 2 and 3, the closed-loop-twist is

And for limb 3 and 1, the closed-loop-twist is

Further, the closed-loop-twist of the reconfigurable base, shown in Fig. 6(b) is

In each decomposed closed-loop-mechanism, only part of all the joints are active, which can be separated from the rest passive joints in the twists given in Eqs. (67) throughout (70) as,

The Eqs. (71)–(74) can be expressed in a matrix form that indicates the relationship between velocity of the active joints and that passive joints as,where \(\dot{\varvec{\theta }}_{a} \;{\text{and}}\;\dot{\varvec{\theta }}_{p}\) denote the velocity vectors of active joints and passive joints as, \(\dot{\varvec{\theta }}_{P} = \left[ {\dot{\theta }_{2} ,\;\dot{\theta }_{3} ,\;\dot{\theta }_{4} ,\;\dot{\theta }_{12} ,\;\dot{\theta }_{13} ,\;\dot{\theta }_{14} ,\;\dot{\theta }_{15} ,\; \ldots \;,\;\dot{\theta }_{35} } \right]^{\text{T}}\), \({\mathbf{J}}_{a} \;{\text{and}}\;{\mathbf{J}}_{p}\) denote the active-Jacobian matrix and passive-Jacobian matrix respectively as

where \({\mathbf{J}}_{pi} = \left[ {\begin{array}{*{20}c} {\varvec{S}_{i2} } & {\varvec{S}_{i3} } & {\varvec{S}_{i4} } & {\varvec{S}_{i5} } \\ \end{array} } \right],\;i = 1\;,\;2\;{\text{and}}\;3\).

The above Jacobian matrixes can be used for singularity and dexterity analysis of the proposed integrated parallel mechanism.