The continuum links utilized in each of the studied planar PCR designs is shown in Figure 1. Throughout this work, we assume that the length of each link is fixed, while the bending can be actively controlled. We further assume, that bending occurs in arcs with a constant curvature, which is a common assumption for a variety of actuation methods (Webster III and Jones, 2010), e.g., utilizing tendons (Rao et al., 2021), multi-backbones (Simaan et al., 2004), push-pull rods (Oliver-Butler et al., 2021), polymer actuators (Moghadam et al., 2015) or pneumatic actuators (Garcia et al., 2020; 2021). We do not consider additional methods of actuation, such as translating or rotating the bases of the continuum links, a practice often done in PCR with passively deforming links. We note that the continuum links throughout this paper are depicted as tendon-driven structures (i.e. including tendons and tendon routing disks). We do this for illustrative purposes only, without loss of generality of the outlined methodology.

We note that while constant curvature assumptions are generally valid in free space, the tip of the continuum links in our designs may be loaded by the weight of the moving platform and by wrenches arising from the coupling between the individual links and end-effector platform. However, the moving platform can be made lightweight in a practical design, such as done by Nuelle et al. (2020), where a constant curvature based kinematic model has been validated for a planar tendon-driven PCR. Moreover, choosing a constant curvature kinematic description is convenient and intuitive as it allows to compare the kinematic properties of the planar 3-DOF PCR with those of rigid planar 3-DOF parallel manipulators (Lilge et al., 2020; Wen and Burgner-Kahrs, 2022).

Figure 1 shows an example continuum link i with fixed length ℓ _i and curvature κ _i . A local frame O _i1 x _i1 y _i1 is attached to the base of the link. The x _i1-axis is parallel to the spacer disk at the link’s base, while the y _i1-axis is orthogonal to it. The trajectory of the link’s distal end is shown in blue. The current position u _i1 of the link’s distal end depends on κ _i and is expressed in its local base frame using constant curvature kinematics (Webster III and Jones, 2010; Wen and Burgner-Kahrs, 2022) u i 1 = 1 κ i 1 − cos ℓ i κ i sin ℓ i κ i . (1) Following this equation, the continuum link is bending in the local positive x-direction for κ _i > 0 and in the local negative x-direction for κ _i < 0. The continuum link is straight and solely extends along the local y-axis. Throughout this paper, we assume that the maximum bending curvature is | κ i | = 2 π ℓ i − 1 , in which case the continuum link forms a closed circle.

The velocity of the link’s distal end position can be obtained by differentiating Eq. 1 with respect to time: u ̇ i 1 = E i u i 1 κ ̇ i , (2) where E i = − 1 κ i ℓ i ℓ i ⁡ cos ℓ i κ i 1 − cos ℓ i κ i − 1 κ i . (3) The velocity vector u ̇ i 1 is tangent to the blue trajectory in Figure 1. In order to describe the orientation of the link’s distal end, an additional frame O _i2 x _i2 y _i2 is attached to its tip. This frame describes the orientation of the link’s distal end with respect to its local base frame O _i1 x _i1 y _i1, where Q _i2 is the rotation matrix between these two frames (Wen and Burgner-Kahrs, 2022) Q i 2 = cos − ℓ i κ i − sin − ℓ i κ i sin − ℓ i κ i cos − ℓ i κ i . (4) The velocity of this orientation can again be obtained by differentiating Eq. 4 with respect to time Q ̇ i 2 = − κ ̇ i ℓ i E Q i 2 , (5) where E = 0 − 1 1 0 . (6)

Both, the linear and angular velocities of the link’s distal end depend on the rate of change in curvature κ ̇ i , which itself depends on the actuator velocity q ̇ i . The relationship between κ ̇ i and q ̇ i is defined by the utilized actuation principle.

Each 3-DOF planar PCR design studied throughout this work consists of three identical legs attached to a moving end-effector platform. Each leg consists of an actuated continuum link in addition to two passive joints, which can either be revolute or prismatic, according to Grübler’s mobility formula (Merlet, 2005). An enumeration of possible planar 3-DOF legs, considering different combinations of the continuum link and passive joints, is shown in Table 2. We are using the following kinematic notation for the planar PCR: R and P respectively stand for a passive revolute joint and a passive prismatic joint, while F stands for an actuated continuum link.

Since the continuum link in each leg is actuated while the revolute and prismatic joints are passive, there are in total 12 planar 3-DOF legs, which is less than the number (21 in total) of rigid planar 3-DOF legs identified by Bonev et al. (2003). Further, following the investigations outlined by Bonev et al. (2003), some of the listed leg designs will lead to uncontrollable linear velocities, if two or more of these legs are used in a 3-DOF PCR, due to the existence of two passive prismatic joints. There are three of such legs which are marked with a superscript ^×. Thus, only the remaining nine legs yield planar 3-DOF symmetrical PCR.

Figure 2 shows the geometric description of a 3-FRR PCR as an example. In order to describe each planar PCR design, a fixed reference frame Oxy is attached to its base and a moving frame O′x′y′ is attached to its moving platform. The vector p and matrix Q respectively represent the position of the origin and the orientation of the moving frame attached to the platform. The rotational angle ϕ defines the moving frame’s rotation with respect to the fixed frame. The individual legs of the PCR are denoted with i ∈ {1, 2, 3}. The curvature of the ith continuum link is described by κ _i . The position vector a _i points to a reference point A _i on the first rigid joint or the continuum link of a leg, which could be the centre of the revolute joint (for 3-RFR, 3-RRF, 3-RFP, and 3-RPF PCRs), a point on the axis of the prismatic joint (for 3-PFR and 3-PRF PCRs), or the proximal endpoint of the continuum link (for 3-FRR, 3-FRP and 3-FPR PCRs). Vector u _i is pointing from one end of the continuum link to the other, while vector v _i is pointing from the start point (or the reference point on the prismatic joint) to the endpoint of the rigid part of a leg. Finally, b _i0 is the position vector pointing to the endpoint B _i of a leg, expressed in the moving frame O′x′y′. Except for vector b _i0, all vectors mentioned above are expressed in the fixed frame Oxy.

A planar 3-RFR PCR is illustrated in Figure 3, where ρ _i stands for the distance between points A _i and B _i . Kinematic modeling of this architecture has been developed in (Lilge et al., 2020; Wen and Burgner-Kahrs, 2022), hence only the expressions of the Jacobian matrices are given here for the sake of clarity. Jacobian matrix J of the PCR is written as J = u 1 T u 1 T E Q b 10 u 2 T u 2 T E Q b 20 u 3 T u 3 T E Q b 30 , (10) where u i = p + Q b i 0 − a i . (11) Here, p and Q are the position and orientation of the moving frame O′x′y′ with respect to the fixed frame Oxy, respectively. Thus, u _i is pointing from A _i to B _i . The rotation matrix E is defined in Eq. 6. Jacobian matrix K can be written as K = diag k 1 k 2 k 3 , (12) where k i = ρ i α i , (13) with ρ i = 2 κ i sin ℓ i κ i 2 , α i = ℓ i κ i cos ℓ i κ i 2 − ρ i κ i . (14) The derivation of these equations can be found in (Wen and Burgner-Kahrs, 2022). Each row in Jacobian matrix J represents a Plücker line passing through points A _i and B _i in the corresponding leg, as indicated by the dashed red lines in Figure 3. This is similar to the case of the rigid planar 3-RRR parallel manipulator studied by Bonev et al. (2003).

A planar 3-FRR PCR is illustrated in Figure 4. Again, the kinematic modeling of this architecture has previously been developed in (Lilge et al., 2020; Wen and Burgner-Kahrs, 2022). The Jacobian matrix J of the PCR is written as J = v 1 T v 1 T E Q b 10 v 2 T v 2 T E Q b 20 v 3 T v 3 T E Q b 30 , (15) where v i = p + Q b i 0 − u i − a i , (16) and u _i is u _i1, as defined in Eq. 1, expressed in the fixed frame Oxy. Here, each row represents a Plücker line passing through the rigid link in the corresponding leg, as indicated by the red dashed lines in Figure 4. This is similar to the case of the rigid planar 3-RRR parallel manipulator studied by Bonev et al. (2003). Jacobian matrix K can be written as K = diag k 1 k 2 k 3 , (17) with k i = v i T Q i 1 E i u i 1 . (18) Here, Q _i1 is the rotation matrix of the local frame O _i1 x _i1 y _i1, which is attached to the base of the ith continuum link, with respect to the fixed frame Oxy.

The geometric description of the 3-RRF PCR is shown in Figure 5. The geometric constraint for each of the legs is that the length of the rigid RR link is constant at any instant, resulting in the following constraint equation p + Q b i 0 − a i − u i T p + Q b i 0 − a i − u i = v i 2 , (19) in which v _i is the Euclidean norm of vector v _i connecting the centre of the first revolute joint to that of the second revolute joint in the ith leg. Differentiating Eq. 19 with respect to time yields v i T p ̇ + ϕ ̇ v i T E Q b i 0 − v i T u ̇ i = 0 , (20) where v i = p + Q b i 0 − a i − u i . (21) Vector u _i can be expressed in the local frame O _i1 x _i1 y _i1 of continuum link i, which is attached to its end connected to the moving platform. The orientation of this local frame with respect to the moving platform frame is constant. We rewrite vector u _i in the following form u i = − Q i 1 u i 1 , (22) where Q _i1 is the rotation matrix of the local frame with respect to the fixed frame Oxy. Vector u _i1 is defined in Eq. 1. Differentiating Eq. 22 with respect to time, one can obtain u ̇ i = − ϕ ̇ E Q i 1 u i 1 − Q i 1 u ̇ i 1 = ϕ ̇ E u i − Q i 1 u ̇ i 1 , (23) in which Q ̇ i 1 = ϕ ̇ E Q i 1 , and u ̇ i 1 is defined in Eq. 2. Substituting Eq. 23 into Eq. 20 and rearranging terms leads to v i T p ̇ + ϕ ̇ v i T E Q b i 0 − u i = − κ ̇ i v i T Q i 1 E i u i 1 . (24) Jacobian matrices J and K of the 3-RRF PCR can then be constructed from Eq. 24. Matrix J is written as J = v 1 T v 1 T E Q b 10 − u 1 v 2 T v 2 T E Q b 20 − u 2 v 3 T v 3 T E Q b 30 − u 3 , (25) in which each row represents a Plücker line passing through the rigid link in the corresponding leg, as indicated by the red dashed lines in Figure 5, and the diagonal elements in matrix K are k i = − v i T Q i 1 E i u i 1 . (26)

A planar 3-RFP PCR is schematically represented in Figure 6. The proximal end of the passive prismatic joint in each leg is rigidly connected to the continuum link, while the distal end of this joint is fixed on the moving platform. In order to establish the constraint equations, we first define a unit vector n _i which is expressed in the fixed frame Oxy and is orthogonal to vector v _i in the same leg. This vector is constant since the direction of vector v _i with respect to the moving frame is maintained at any instant. The orthogonality between vectors n _i and v _i in each leg can be written as n i T v i = 0 , (27) where vector v _i is expressed in the same form as that shown in Eq. 21. To make the following derivations more intuitive, we rewrite the constraint equation as Q n i 0 T v i = 0 , (28) where n _i0 is the unit vector n _i expressed in the moving frame O′x′y′. The time derivative of Eq. 28 yields ϕ ̇ E Q n i 0 T v i + Q n i 0 T v ̇ i = 0 , (29) with v ̇ i = p ̇ + ϕ ̇ E Q b i 0 − u ̇ i . (30)

Similarly to the local frame defined in Figure 5, a local frame O _i1 x _i1 y _i1 is attached to the distal end of the ith continuum link of the 3-RFP PCR (see Figure 6). The orientation matrix of this frame relative to the fixed frame is denoted by Q _i1, which is a function of angle ϕ, i.e., the rotational angle of the moving frame with respect to the fixed frame. This is because the orientation of this local frame relative to the moving platform frame is constant, regardless of the prismatic joint values in the same leg. The time derivative of vector u _i can be derived in the same way as in the previous design and results in Eq. 23.

Substituting the expression of u ̇ i into Eq. 30 and then into Eq. 29 and noting that E ^T = −E, the left summand of Eq. 29, which contains ϕ ̇ , can be rewritten as ϕ ̇ E Q n i 0 T v i + ϕ ̇ Q n i 0 T E Q b i 0 − ϕ ̇ Q n i 0 T E u i = ϕ ̇ Q n i 0 T E Q b i 0 − v i − u i = ϕ ̇ h i , (31) in which h i = Q n i 0 T E Q b i 0 − v i − u i . (32) Hence Eq. 29 can be rewritten into the following form Q n i 0 T p ̇ + ϕ ̇ h i = − κ ̇ i Q n i 0 T Q i 1 E i u i 1 . (33) The Jacobian matrices J and K are constructed from Eq. 33, where matrix J is written as J = Q n 10 T h 1 Q n 20 T h 2 Q n 30 T h 3 = n 1 T n 1 T E a 1 − p 1 n 2 T n 2 T E a 2 − p 2 n 3 T n 3 T E a 3 − p 3 , (34) in which each row represents a Plücker line which is orthogonal to vector v _i and passes through the centre of the revolute joint in the ith leg, as indicated by the red dashed lines in Figure 6. The diagonal elements of matrix K are k i = − Q n i 0 T Q i 1 E i u i 1 = − n i T Q i 1 E i u i 1 . (35)

It should be noted that n _i becomes undefined, when v _i becomes a zero vector. Thus, type I singularities occur in this case, in addition to all other cases in which k _i = 0 (for i = 1, 2 or 3). The same is true for the other following planar PCR design which contain passive prismatic joints.

A planar 3-PFR PCR is schematically represented in Figure 7. The constraint equations of the planar 3-PFR PCR are written as n i T v i = 0 , (36) where n _i is a unit vector expressed in the fixed frame which is orthogonal to vector v _i . Vector n _i is constant with respect to time, as its direction does not change regardless of the value of the prismatic joint in the same leg. Vector v _i is expressed in the same form as that shown in (21). Vector u _i , which is used to express vector v _i , is defined as u _i = Q _i1 u _i1, where Q _i1 stands for the constant orientation matrix of the local frame O _i1 x _i1 y _i1 relative to the fixed frame, and vector u _i1 is expressed in the local frame O _i1 x _i1 y _i1 using Eq. 1.

The time derivative of Eq. 36 leads to n i T v ̇ i = n i T p ̇ i + ϕ ̇ E Q b i 0 − Q i 1 u ̇ i 1 = 0 . (37) Using Eq. 2, we can further write n i T p ̇ + ϕ ̇ E Q b i 0 − κ ̇ i Q i 1 E i u i 1 = 0 , (38) n i T p ̇ + ϕ ̇ E Q b i 0 = κ ̇ i n i T Q i 1 E i u i 1 , (39) where E _i is defined in Eq. 3. Using Eq. 39, Jacobian matrix J can be written in the following form J = n 1 T n 1 T E Q b 10 n 2 T n 2 T E Q b 20 n 3 T n 3 T E Q b 30 . (40) The Plücker line in each leg is orthogonal to vector v _i and passes through the centre of the revolute joint, as indicated by the red dashed lines in Figure 7. The diagonal elements in Jacobian matrix K are k i = n i T Q i 1 E i u i 1 . (41)

A planar 3-PRF PCR is schematically represented in Figure 8. The derivation is similar to the one of the previous design, thus the constraint equations are n i T v i = 0 , (42) where n _i is a unit vector and v _i is defined in Eq. 21. This time, vector u _i is defined as u _i = −Q _i1 u _i1, where Q _i1 stands for local frame O _i1 x _i1 y _i1 relative to the fixed frame, and vector u _i1 is expressed in the local frame O _i1 x _i1 y _i1 using Eq. 1. We note that Q _i1 is not constant with respect to time here, as it moves with the moving platform. However, the direction of n _i does not change with respect to time and is constant for any joint value of the prismatic joint. In total, the time derivative of the constraint equation is obtained as n i T p ̇ + ϕ ̇ E Q b i 0 − ϕ ̇ E u i + κ ̇ i Q i 1 E i u i 1 = 0 , (43) again making use of Eq. 23 and E _i is again defined in Eq. 3. Based on Eq. 43, we can identify J = n 1 T n 1 T E Q b 10 − u 1 n 2 T n 2 T E Q b 20 − u 2 n 3 T n 3 T E Q b 30 − u 3 . (44) The Plücker line in each leg is orthogonal to vector v _i and passes through the centre of the revolute joint, as indicated by the red dashed lines in Figure 8. The diagonal elements of Jacobian matrix K result in k i = − n i T Q i 1 E i u i 1 . (45)

Similar to the previously discussed designs that feature passive prismatic joints, it can be readily observed that the constraint equations of the planar 3-FRP PCR (see Figure 9) are the orthogonality between the unit vector n _i and vector v _i such that n i T v i = 0 . Again, v _i is defined in Eq. 21, in which vector u _i is defined as u _i = Q _i1 u _i1, where Q _i1 stands for local frame O _i1 x _i1 y _i1 relative to the fixed frame, and vector u _i1 is expressed in the local frame O _i1 x _i1 y _i1 using Eq. 1. Skipping some steps, which have been outlined throughout the previous designs, the time derivation of the constraint equations results in n ̇ i T v i + n i T v ̇ i = 0 , (46) ϕ ̇ E n i T v i + n i T p ̇ + ϕ ̇ E Q b i 0 − κ ̇ i Q i 1 E i u i 1 = 0 , (47) n i T p ̇ + ϕ ̇ n i T E Q b i 0 − v i = κ ̇ i n i T Q i 1 E i u i 1 , (48) where E _i is again defined in and Eq. 3. Now the Jacobian matrix J can be obtained as J = n 1 T n 1 T E Q b 10 − v 1 n 2 T n 2 T E Q b 20 − v 2 n 3 T n 3 T E Q b 30 − v 3 . (49) The Plücker line in each leg is orthogonal to vector v _i and passes through the centre of the revolute joint, as indicated by the red dashed lines in Figure 9. The diagonal elements of the Jacobian matrix K consist of k i = n i T Q i 1 E i u i 1 . (50)

A planar 3-RPF PCR is schematically represented in Figure 10. Two local frames, noted O _i1 x _i1 y _i1 and O _i2 x _i2 y _i2, are respectively attached to the distal and proximal ends of the continuum links. The orientation of the former local frame relative to the moving platform frame is constant. In each leg, the unit vector n _i , which is expressed in the moving platform frame, is orthogonal to vector v _i .

Similar to the previously discussed designs, the constraint equations are n i T v i = 0 , (51) where v _i is the same as in Eq. 21 and u _i is defined as u i = − Q i 1 u i 1 , (52) in which Q _i1 is the constant orientation matrix of the local frame O _i1 x _i1 y _i1 with respect to the fixed frame, and u _i1 is defined in the local frame O _i1 x _i1 y _i1 as expressed in Eq. 1. The time derivative of the constraint equations yields n ̇ i T v i + n i T v ̇ i = 0 , (53) with v ̇ i = p ̇ + ϕ ̇ E Q b i 0 − u ̇ i = p ̇ + ϕ ̇ E Q b i 0 − ϕ ̇ E u i + κ ̇ i Q i 1 E i u i 1 , (54) where the expression for u ̇ i can be derived based on the procedure given in Eq. 22 and Eq. 23, E _i has been defined in Eq. 3. In order to calculate the time derivative of vector n _i , we rewrite this vector in the following form n i = Q i 1 Q i 2 n i 2 , (55) where Q _i2 is the orientation matrix of the local frame O _i2 x _i2 y _i2 relative to the local frame O _i1 x _i1 y _i1 as defined in Eq. 4. Unit vector n _i2 denotes unit vector n _i when expressed in the local frame O _i2 x _i2 y _i2. The direction of the unit vector n _i2 is constant with respect to time since the geometric relationship between the prismatic joint and the proximal end of the continuum link in the same leg remains unchanged. The time derivative of vector n _i can then be written as n ̇ i = Q ̇ i 1 Q i 2 n i 2 + Q i 1 Q ̇ i 2 n i 2 . (56) Since the local frame is attached to the moving frame, its derivation with respect to time depends on the angular velocity of the moving platform such that Q ̇ i 1 = ϕ ̇ E Q i 1 . Together with the time derivative Q ̇ i 2 from Eq. 5, we obtain. n ̇ i = ϕ ̇ E Q i 1 Q i 2 n i 2 − κ ̇ i ℓ i Q i 1 E Q i 2 n i 2 (57) = ϕ ̇ E n i − κ ̇ i ℓ i E n i . (58)

Substituting Eq. 54 and Eq. 58 into Eq. 53 and rearranging terms, and noting that E ^T = −E one can obtain. n i T p ̇ + ϕ ̇ n i T E Q b i 0 + ϕ ̇ E n i T v i − ϕ ̇ n i T E u i = κ ̇ i k i , (59) n i T p ̇ + ϕ ̇ n i T E Q b i 0 − ϕ ̇ n i T E v i − ϕ ̇ n i T E u i = κ ̇ i k i , (60) n i T p ̇ + ϕ ̇ n i T E Q b i 0 − v i − u i = κ ̇ i k i , (61) where k i = − n i T Q i 1 E i u i 1 + ℓ i E n i T v i , (62) = − n i T Q i 1 E i u i 1 − ℓ i n i T E v i , (63) = − n i T Q i 1 E i u i 1 + ℓ i E v i . (64) The Jacobian matrices now result from Eq. 61 in J = n 1 T n 1 T E Q b 10 − v 1 − u 1 n 2 T n 2 T E Q b 20 − v 2 − u 2 n 3 T n 3 T E Q b 30 − v 3 − u 3 = n 1 T n 1 T E a 1 − p n 2 T n 2 T E a 2 − p n 3 T n 3 T E a 3 − p , (65) and K = diag[k ₁ k ₂ k ₃], where the expression for k _i are given in Eq. 64. The Plücker line in each leg is orthogonal to vector v _i and passes through the centre of the revolute joint, as indicated by the red dashed lines in Figure 10.

Each of the legs of the 3-FPR PCR contains a passive prismatic joint, indicating that the geometric constraint is the orthogonality between vectors n _i and v _i in the same leg n i T v i = 0 . (66) Before taking the time derivative of the constraint equations, we define vector n _i , which is expressed in the fixed frame, in the following form n i = Q i 1 Q i 2 n i 2 , i = 1,2,3 (67) in which matrix Q _i1 represents the constant orientation of the local frame O _i1 x _i1 y _i1 with respect to the fixed frame, matrix Q _i2 has the same form as that given in Eq. 4, and the constant unit vector n _i2 is expressed in the local frame O _i2 x _i2 y _i2. The time derivative of the constraint equations yields n ̇ i T v i + n i T v ̇ i = 0 . (68) Noting that both Q _i1 and n _i2 in Eq. 67 are constant with respect to time, as frame O _i1 x _i1 y _i1 is fixed to the corresponding leg’s base and the direction of n _i2 remains the same with respect to frame O _i2 x _i2 y _i2, n ̇ i can be written as n ̇ i = − κ ̇ i ℓ i Q i 1 E Q i 2 n i 2 = − κ ̇ i ℓ i E n i . (69) Further, v ̇ i is defined as v ̇ i = p ̇ + ϕ ̇ E Q b i 0 − u ̇ i . (70) In this design u _i = Q _i1 u _i1, where vector u _i1 is expressed in the local frame O _i1 x _i1 y _i1 using Eq. 1, while its derivative is given in Eq. 2. Thus, we can further write v ̇ i = p ̇ + ϕ ̇ E Q b i 0 − κ ̇ i Q i 1 E i u i 1 . (71) Substituting Eq. 69 and Eq. 71 into Eq. 68 and rearranging terms in a similar way than for the previous design, one can obtain n i T p ̇ + ϕ ̇ n i T E Q b i 0 = κ ̇ i k i , (72) in which the diagonal elements in Jacobian matrix K are k i = n i T Q i 1 E i u i 1 − ℓ i E v i , (73) and Jacobian matrix J is expressed as J = n 1 T n 1 T E Q b 10 n 2 T n 2 T E Q b 20 n 3 T n 3 T E Q b 30 . (74) The Plücker line in each leg is orthogonal to vector v _i and passes through the centre of the revolute joint, as indicated by the red dashed lines in Figure 11.

In (Bonev et al., 2003), the concept of reciprocal screws has been utilized to kinematically model planar PKMs. Although the velocity equations of the planar PCRs in this study are not developed based on this concept, the Jacobian matrices J have a similar physical meaning like the corresponding matrices in (Bonev et al., 2003). Therefore, type II singularity analysis of planar PCRs and their rigid counterparts can be carried out using the same framework, such as Grassmann line geometry (Wen and Burgner-Kahrs, 2022).

As pointed out in the previous section, each of the rows in Jacobian matrix J of a planar PCR can be represented by a Plücker line. Based on our findings, such a Plücker line can be geometrically identified by the following rules:

1) If a leg of the planar PCR contains two passive revolute joints, the Plücker line passes through the centres of both joints;

Generally, Jacobian matrix J becomes singular and type II singularities occur when its three Plücker lines intersect at a common point or are parallel to each other. The loci and occurrences of these singularities depend on both the current joint configuration of the PCR, but also on its design parameters and the leg assembly. Example type II singularity occurrences for three of the considered 3-DOF PCR, namely the 3-RFR, 3-FRR and 3-PFR designs, are shown in Figure 12. The occurrences of II singularities for the other designs can be obtained analogously by investigating the corresponding Plücker lines of J.

It is noted, that the Plücker lines for the 3-RFP, 3-PFR, 3-PRF, and 3-FRP PCRs can never be parallel as long as the prismatic joints are not aligned. Due to this, some of these designs might in fact be free of type II singularities, if their Plücker lines additionally never intersect. For instance, type II singularities cannot occur in the 3-RFP and 3-FRP PCRs, since their Plücker lines will never intersect in praxis due to mechanical interferences.

It can be concluded that type II singularity analysis is very straightforward, as the Plücker lines can be identified in a geometrically intuitive way. It is further apparent that the loci and occurrences of type II singularities are mostly analogous to conventional PKM, which usually have Jacobian matrices J consisting of similar Plücker lines. However, throughout the next section we will show, that the type I singularity occurences for the PCR studied in this work are quite different from conventional PKM and are less intuitive to understand.

Type I singularities of rigid PKMs generally occur when either an actuator (usually a linear actuator) reaches its extremes of extension or when the determinant of the Jacobian matrix K is equal to 0. While the first case is usually known directly from the resulting joint values of a design, the second case can often be identified in a straightforward manner by observing the configuration and resulting geometry of the manipulator. However, due to the bending motions of the continuum links, this geometric method might not be intuitive for PCRs. To elaborate on this difference, a comparison study of a rigid 3-RRP parallel architecture and a 3-FRP PCR is provided in the following. Both designs, which are exhibiting a similar kinematic structure, are depicted in Figure 13.

The Jacobian matrix K of the conventional 3-RRP robot can be expressed as K = n 1 T E u 1 0 0 0 n 2 T E u 2 0 0 0 n 3 T E u 3 , (75) where E is a 90-degree rotation as defined in Eq. 6. The diagonal entries of the corresponding matrix for the continuous counterpart, i.e., the 3-FRP robot, are given in Eq. 50, such that matrix K results in K = n 1 T Q 11 E 1 u 11 0 0 0 n 2 T Q 21 E 2 u 21 0 0 0 n 3 T Q 31 E 3 u 31 , (76) where Q _i1 is the orientation of the base of continuum link in leg i, E _i is defined in Eq. 3 and u _i1 is vector u _i expressed in the local frame O _i1 x _i1 y _i1 as defined in Eq. 1.

It can be observed from Eq. 75 that a type I singularity occurs when vectors n _i and u _i (with i = 1, 2, 3) are aligned, i.e., the distal link is perpendicular to the proximal link in the same leg (Bonev et al., 2003), resulting in det(K) = 0. Thus the occurrences of type I singularities can always be associated with the orientation of these two links, allowing to intuitively detect them based on geometry.

On the contrary, the determinant of Jacobian matrix K for the 3-FRP PCR becomes 0 when vector n _i is orthogonal to vector Q _i1 E _i u _i1 (with i = 1, 2, 3). However, unlike for its rigid counterpart, the direction of the latter vector, which is different from vector Eu _i in the 3-RRP rigid parallel mechanism, is not constantly associated with a link. In fact, its direction is always tangent to the course of the tip of the corresponding continuum link, which is visualized in Figure 1. Thus, the orthogonality between these two vectors is not intuitively observed from the configurations of the PCR.

Throughout the remainder of this section, we are providing additional insights into the occurrence of type I singularities for each PCR design discussed in this paper. We are further investigating the existence of different possible postures of the legs in each design for a given desired end-effector pose. This concept has first been discussed for conventional planar parallel robots by Chablat and Wenger (1998), who further introduce the idea of working modes that are separating sets of different leg postures. We show, that type I singularities of the PCR discussed in this paper separate different leg posture solutions. Thus, type I singularities occur, when the number of possible leg posture solution varies, e.g., when the number of possible solutions changes from two to one or from four to two. In this case, two separate working modes of the PCR join. We note that an analogous behaviour was shown for conventional PKM by Chablat and Wenger (1998) and was conjectured for passive link PCR by Briot and Goldsztejn (2021).

The results of this section are depicted in Figure 14. For each design, we show possible multiple leg postures for one leg of the corresponding design, connecting the leg’s base A _i to the platform coupling point B _i . The shown solutions meet the geometrical coupling constraints, which depend on possible positions of the end of the continuum link, shown in a blue solid line, and the possible motions allowed from the passive joint, shown in red dashed line. The figure further shows example cases, in which type I singularities for an individual leg in a design exist. As stated above, the number of possible leg postures changes in these configurations. Additionally, the tangents of the dashed red line and solid blue line at the corresponding solution become parallel for most designs in this case. We believe that the insights gained in this section can lead to a better geometrical understanding of type I singularity occurrences, which can for instance be useful to determine the designs workspace boundaries, which are usually defined by such.

We note that observing the results from the previous section, it can be seen that designs that are featuring the same kinematic structure in their legs, but with a reversed arrangement, have similar type I singularity conditions. For instance, the 3-FRR and 3-RRF designs have type I singularities when Eq. 18 and Eq. 26 are equal to 0 for any i. It can be seen that these equations only differ in their sign, leading to the same conditions for type I singularities. The same is true for most of the remaining designs, with the 3-RFR being the exception. Thus, through the following we will only discuss one of each of these designs, while the type I singularity occurrences of the corresponding other design can be obtained analogously.