Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula

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Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula Genliang Chen a,b,∗, Lingyu Kong b, Qinchuan Li c, Hao Wang a,b, Zhongqin Lin a,b a

State Key Laboratory of Mechanical Systems and Vibration, Shanghai Jiao Tong University, Shanghai, 200240, China Shanghai Key Laboratory of Digital Manufacturing for Thin-Walled Structures, Shanghai Jiao Tong University, Shanghai 200240, China c The Mechatronic Institute, Zhejiang Sci-Tech University, Hangzhou 310018, China b

a r t i c l e

i n f o

Article history: Received 5 June 2017 Revised 13 October 2017 Accepted 9 November 2017 Available online xxx Keywords: Kinematic calibration Error modeling Parallel manipulators Kinematic-parameter identification POE formula

a b s t r a c t Determination of the identifiable parameters plays an important role in kinematic calibration of robot manipulators. For the conventional serial robots, a consensus has been reached on this problem. However, for the parallel manipulators, there is a lack of effective methods to analyze the parameters’ identifiability. This paper presents a systematic approach to establish complete, minimal and continuous error models for kinematic calibration of parallel manipulators. Using the product of exponentials (POE) formula, orthogonal partitioning matrices can be constructed in a straightforward way to determine and eliminate the redundant error components. Hence, error models satisfying completeness, minimality and continuity requirements, can be established for the parallel manipulators. The result shows that the maximal number of identifiable parameters is 4r + 2 p + 6 in a parallel manipulator, where r and p denote the quantities of equivalent revolute and prismatic joints, respectively. And this number is independent of the topology of parallel manipulators, such as the number of limbs, the assignments of actuations, and the geometric constraints within/between the limbs. Furthermore, this conclusion is in agreement with the well-known one for serial robots. Hence, the maximal number of identifiable parameters for both serial and parallel manipulators can be unified by one unique formula. © 2017 Published by Elsevier Ltd.

1. Introduction Kinematic calibration serves as an economical and effective method to improve the absolute positioning accuracy of robot manipulators [1–5]. In order to guarantee the computational stability and efficiency during the iterative identification of the kinematic parameters, three basic requirements, namely completeness, continuity and minimality [6,7], should be satisfied by the calibration models. Although these requirements are proposed for the kinematic models, it is easier to understand to interpret them into those for the corresponding error models as

∗ Corresponding author at: State Key Laboratory of Mechanical Systems and Vibration, Shanghai Jiao Tong University, Building of Mechanical Engineering, Room A-611, DongChuan Road 800, Shanghai, 200240, China. E-mail address: [email protected] (G. Chen).

https://doi.org/10.1016/j.mechmachtheory.2017.11.003 0094-114X/© 2017 Published by Elsevier Ltd.

Please cite this article as: G. Chen et al., Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula, Mechanism and Machine Theory (2017), https://doi.org/10.1016/j.mechmachtheory.2017.11.003

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Completeness: A complete error model must have enough parameters to describe any possible geometric deviations which influence the pose of the robot’s end-effector. Continuity: Infinitesimal changes in the geometric structure of the robot should result in infinitesimal values for the error parameters in the calibration models. Minimality: There is no redundant parameters in the error models. Mathematically, the errors must be independent of each other, meanwhile the corresponding identification matrix should be column full rank.

For the conventional serial robots, the issues of kinematic calibration has be intensively studied during the past decades. Comparing with other calibration models, such as the Denavit–Hartenberg convention based ones [8–10], the zero-reference position model [11], and the complete and parametrically continuous (CPC) models [12], error models based on the POE formula have inherent advantages because of the smooth property of the exponential map from the Lie algebra se(3) to the Lie group SE(3). First, the kinematic parameters in the POE based calibration models vary smoothly with the changes in the joint axes, which guarantees that the error models are singularity-free. Second, because any rigid-body motion can be regarded as a screw one which corresponds to a certain twist, these error models are also complete. In addition, in POE formulation, all joint axes are described based on line geometry. So it can be uniformly applied to robot manipulators with both revolute and prismatic joints. Because of the above merits, since Okamura and Park [13] first introduced the POE formula to the robot kinematic calibration in 1996, it has attracted more and more attention in the robotics community during the past years [14–19]. As mentioned in the above, the calibration models based on the POE formulation are naturally complete and continuous. Therefore, the minimal requirement, namely the identifiability condition for the error parameters, becomes the main issue for the reliable and stable identification of the robot manipulators’ kinematic parameters [7]. Lou et al. [15] indicated that the joint offset errors are not identifiable by using the original POE calibration models [13]. Then, they proposed an improved formulation in which both joint offset errors and joint twist constraints can be eliminated, such that the parameter identification process can be reduced into unconstrained optimization problems. By introducing an explicit expression of the derivatives in SE(3), He et al. [16] proposed a general POE based error modeling approach for the kinematic calibration of serial robots. However, since the joint twist constraints to the error components are neglected, extra normalization and orthogonalization deductions are required during the identification process of the manipulators’ kinematic parameters, which indicates that the error model can not satisfy the requirement of minimality. Recently, Wu et al. [17] established a minimal POE based error model for the kinematic calibration of serial robots by using the singularity-free representation of line geometry. Apart from the original POE formula, Chen et al. [14] also proposed an alternative presentation which is referred to as the local POE model for the kinematic calibration of serial robots. In this formulation, all joint twists are represented as constants without any errors in their corresponding local frames. Hence, kinematic errors of the robot manipulators only exist in the initial poses of the consecutive local frames, which are also represented in the exponential maps from se(3) to SE(3). The main advantage of the local POE based calibration model is that the pose error of the robot’s tool frame is accumulated by the kinematic ones of the links in a successive manner, which makes it concise and easy to understand. By a particular choice of the links’ local frames, Chen et al. [18] presented a reformulation to the local POE formulation, such that the joint offset errors can be integrated into the geometric errors of the links. They systematically investigated the identifiability of parameters in serial robot calibration and proved that the maximum number of identifiable parameters is 4r + 2 p + 6 which is independent of the formulation methods used for the kinematic modeling. Furthermore, analytical approaches were proposed to identify and eliminate the redundant error parameters in both local and global POE formulas. So far, a consensus has been reached on the identifiability of kinematic parameters in serial robot calibration. Different methods [17,18] can be used to establish complete, continuous and minimal error models for the kinematic calibration. However, for parallel manipulators, although a lot of research [20–29] has been conducted on their kinematic calibration, investigations on the completeness, continuity and minimality property of the calibration models are relatively few [30–34]. Instead, most of the existing literatures focused more on the measurement phase during the calibration process, such as the concept of self-calibration [21–23], several new sensoring techniques [25,26] and optimal selection of the measurement configurations [27,28,35]. Due to the complexity of forward kinematics, error models for the kinematic calibration of the parallel manipulators are usually derived based on their inverse kinematics [27,28,36]. Therefore, some imperfect geometric errors, such as the axes offset in the universal joints [37], can not be taken into account in the system calibration models. Hence, these error models do not satisfy the completeness requirement from the calibration point of view. On the other hand, several general approaches [38–40] have been established for the geometric error modeling of parallel manipulators. These models have the abilities of containing all of the potential error parameters, but the unidentifiable error parameters still need to be further investigated. Otherwise, Lin [30] proposed an empirical formula 3r + p + SS + E + 6 to predict the maximum number of the identifiable parameters in parallel manipulators, where SS is the number of links which are connected by two spherical joints and E is the number of joint sensors. Then, Vischer [31] and Legnani et al. [33] reformulated this formula by taking the kinematic loops and the cylindrical joints into account, respectively. However, both of the equations were proposed mainly based Please cite this article as: G. Chen et al., Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula, Mechanism and Machine Theory (2017), https://doi.org/10.1016/j.mechmachtheory.2017.11.003

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Fig. 1. Local POE formulation of a general kinematic limb isolated from the parallel manipulator.

on experience and no rigorous proof has been advanced to validate the correctness of such kind of conclusions to the identifiability analysis of parallel manipulators. In this paper, a systematic approach is presented to establish the complete, minimal and continuous error models for the kinematic calibration of parallel manipulators. In the proposed method, all multi-DOF joints are equivalently replaced by the combination of revolute and prismatic ones. Thus, the method developed for serial robots [18] can be extended to the error modeling of the individual limbs in parallel manipulators. The influence of passive joints on the identifiability of kinematic parameters is studied based upon the reciprocal screw theory. Then, an orthogonal partitioning matrix can be constructed in a straightforward way to identify and eliminate the redundant error parameters due to the existence of passive joints. Finally, calibration models only comprising independent errors can be obtained for the kinematic-parameter identification of parallel manipulators. It has been proved that the maximum number of identifiable parameters is 4r + 2 p + 6 in a nonoverconstrained parallel manipulator which equivalently possesses r revolute and p prismatic joints, respectively. This result is independent of the manipulators’ concrete topologies, such as the number of limbs, the placement of actuations and the geometric constraint between the limbs. Furthermore, this conclusion is in accordance with the consensus for the serial robots. As a result, the maximal number of identifiable parameters for both serial and parallel manipulators can be uniformly represented by one identical formula. Accordingly, a general calibration strategy is developed for kinematic-parameter identification of parallel manipulators based on the proposed error modeling method. This paper is organized as follows: In Section 2, a general method is presented for the error modeling of parallel manipulators based on the POE formula. Thereafter, identifiability analysis of the obtained error models is conducted in Section 3, where linear orthogonal operators are constructed to determine and eliminate the redundant errors induced by the passive joints. Accordingly, a general strategy is proposed in Section 4 for parameter identification of parallel manipulators by using the inverse kinematics and forward error models. Finally, conclusions are drawn in Section 5.

2. Error modeling of parallel manipulators In this section, a general method for the error modeling of parallel manipulators is presented based upon the local POE formulation of the limbs’ forward kinematics. As indicated above, the calibration method based on the POE formula has been well established for serial robots. Therefore, some mathematical fundamentals about the screw theory and POE formula are provided in Appendix A for reference. It is assumed that the inverse position analysis of the parallel manipulator can be accomplished in a straightforward manner. Then, all joint variables of the manipulator’s limbs, including the passive ones, can be obtained according to the pose of end-effector via the inverse kinematics as

θi = f IK, i (gst ), i = 1, · · · , n

(1)

where gst ∈ SE (3 ) denotes the pose of the tool frame {T} with respect to the spatial one {S}, as shown in Fig. 1. θi = [θi,1 , · · · , θi, fi ]T ∈ R fi ×1 is the vector of the limb’s joint variables, where fi represent the number of equivalent 1-DOF joints in the limb. And n corresponds to the number of limbs in the parallel manipulator. It should be noted that in the proposed method all multiple DOF joints in the parallel manipulator are regarded as the combinations of 1-DOF ones. Hence, the universal joints are replaced by two revolute ones with concurrent axes. While the spherical joints are constructed by three revolute ones whose axes are intersected at a common point in an orthogonal way. Please cite this article as: G. Chen et al., Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula, Mechanism and Machine Theory (2017), https://doi.org/10.1016/j.mechmachtheory.2017.11.003

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2.1. Limbs’ error models based on the local POE formula Generally, the reason that the inverse kinematics of parallel manipulators can be derived in closed-form is because of some particular geometric constraints between the joint axes within the limbs, such as parallel, perpendicular and concurrent conditions. As a consequence, the inverse displacement solutions (1) only works under the manipulators’ nominal kinematic parameters. When the links’ geometric errors are taken into account, the limbs become general serial kinematic chains due to the violation of geometric constraints. Usually, no analytical solutions can be achieved any more for such kind of inverse kinematics problems. Thus, precise error models can not be directly obtained by means of differentiating the nominal inverse kinematics of limbs. Therefore, the local representation of POE formula [14,18] for the forward kinematics of serial robots is employed in this paper for the error modeling of the individual limbs of parallel manipulators. The concept of local POE model was initially proposed by Chen et al. [14] for the kinematic calibration of serial robots. Comparing with the global one, the advantage of this approach is its easiness in relating to the conventional error modeling methods using consecutive homogeneous transformation matrices (HTM). A reformulation to this method has been developed in our previous work [18], such that the identifiable parameters in kinematic calibration of serial robots can be determined analytically. In this section, a general kinematic limb, as illustrated in Fig. 1, is provided as an example to demonstrate the procedures of error modeling using local POE formula. As shown in the figure, two local coordinate frames, termed {Ri, j } and {Fi, j }, are constructed for   each link at their proximal and distal ends, respectively. The coordinate axes of these frames are denoted by xi, j , yi, j , zi, j and ui, j , vi, j , wi, j , in which zi, j and wi, j keep coincident with joint j − 1 and j, respectively. It is assumed





that, at the home configuration, the corresponding distal and proximal frames, namely Fi, j−1 and {Ri, j }, are coincident with each other. It should be noted that the first proximal frame {Ri, 0 } is coincident with the spatial one {S} and the last   distal one Fi, fi is just the tool frame {T}. Then, the relative poses between any two consecutive local frames can be represented as

gi, j = exp(pˆ i, j ), Ti,j j−1 = exp(ζˆi, j θi, j )

(2)

where gi, j ∈ SE (3 ), j = 0, · · · , fi denotes the homogeneous transformation matrix of {Ri, j } with respect to {Fi, j } which is uniquely determined by the geometry of link j. The twist coordinates (pˆ i, j )∨ = pi, j ∈ R6×1 can be regarded as the link’s



j

kinematic parameters corresponding to gi, j . Meanwhile, Ti, j−1 ∈ SE (3 ) represents the HTM of Fi, j−1

ζˆi, j ∈ se(3 ) is the joint twist represented in



Fi, j−1





with respect to {Ri, j }.

and θ i, j denotes the joint displacement as defined in (1). Since the

shafts of joints keep coincident with the coordinate axes of local frames, the twist ζ i, j yields to [e3 , 03 ]T and [03 , e3 ]T for revolute and prismatic joints, respectively. Here, e3 = [0, 0, 1]T is the unit vector of z-axis in local frames and 03 = [0, 0, 0]T is the 3 × 1 zero vector. Therefore, in limb i, the end-effector pose can be represented by the concatenation of all those homogeneous transformation matrices between the consecutive local frames as

gi,st (θi ) = gi,0 T1i,0 · · · gi, fi −1 Ti,fi f −1 gi, fi = gi,0 exp(ζˆi,1 θi,1 ) · · · gi, fi −1 exp(ζˆi, fi θi, fi ) gi, fi

(3)

i

By differentiating the forward kinematics (3) with respect to {S}, the pose error of the end-effector can be obtained as the twist error of {T} with respect to {S} as fi fi   ∨      δ yi = δ gi,st g−1 = Ad gi,0 T1i,0 · · · gi, j−1 Ti,j j−1 Ai, j δ pi, j + Ad gi,0 T1i,0 · · · gi, j−1 ζi, j δθi, j i,st j=0

(4)

j=0

where (δ gi, j g−1 )∨ = Ai, j δ pi, j denotes the twist error of {Fi, j } with respect to {Ri, j }, which can be related to the link’s kinei, j matic errors δ pi, j through the 6 × 6 matrix

Ai, j = I6 +

4−ϕi, j Sϕi, j −4Cϕi, j 2ϕi,2 j

Zi, j +

4ϕi, j −5Sϕi, j +ϕi, j Cϕi, j 2ϕi,3 j

Z2i, j +

2−ϕi, j Sϕi, j −2Cϕi, j 2ϕi,4 j

Z3i, j +

2ϕi, j −3Sϕi, j +ϕi, j Cϕi, j 2ϕi,5 j

Z4i, j

where Zi, j = ad(pˆ i, j ) ∈ R6×6 denotes the adjoint representation of pˆ i, j ∈ se(3 ) and ϕ i, j is its magnitude, as defined in (38) (For more details, refer [41]). Here, Cϕi, j and Sϕi, j are abbreviations for cos (ϕ i, j ) and sin (ϕ i, j ), respectively. Since the  joint twists ζ i, j are represented in their local frames as invariant constants, the twist error of {Ri, j } with respect to Fi, j−1 can be derived in terms of the joint’s displacement error as



δ Ti,j j−1 (Ti,j j−1 )−1

∨

= (ζˆi, j δθi, j )∨ =

ζi, j δθi, j

(5)

In the error model (4), the adjoint transformation is used to mapped the twist errors from their local frame to the spatial one. And the relative pose of {Ri, j } with respect to {S} can be rewritten as

Ti,j 0 = gi,0 T1i,0 · · · gi, j−1 Ti,j j−1 = exp(ξˆi,1 θi,1 )exp(ξˆi,2 θi,2 ) · · · exp(ξˆi, j θi, j ) gi,0 · · · gi, j−1

(6)

where ξˆi, j = gi,0 · · · gi, j−1 ζˆi, j g−1 · · · g−1 denotes the coordinates of joint twist j in {S}, which can be represented in vector i, j−1 i,0 forms as ξi, j = Ad(gi,0 · · · gi, j−1 ) ζi, j according to (36).

Please cite this article as: G. Chen et al., Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula, Mechanism and Machine Theory (2017), https://doi.org/10.1016/j.mechmachtheory.2017.11.003

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Fig. 2. Transformation of the links’ kinematic errors from their local frames to the spatial one.

Substituting (6) into (4), the pose error of the limb’s end-effector can be rewritten in a matrix form as

δ yi = i i δ pi + i i δθi where δ pi =

[δ pTi,0 , δ pTi,1 , · · ·

, δ pTi, f ]T i

(7) ∈

R6( fi +1 )×1

is the vector of limb’s kinematic errors. δθi = [δθi,1 , · · · , δθi, fi

]T

∈

R fi ×1

is

the error vector of the joint variables. The coefficient matrices are given by

  

i = I6 , Ad exp(ξˆi,1 θi,1 ) , · · · , Ad exp(ξˆi,1 θi,1 ) · · · exp(ξˆi, fi θi, fi ) ∈ R6×6( fi +1)     i = Blockdiag Ai,0 , Ad(gi,0 )Ai,1 , · · · , Ad gi,0 · · · gi, fi −1 Ai, fi ∈ R6( fi +1)×6( fi +1)   

i = I6 , Ad exp(ξˆi,1 θi,1 ) , · · · , Ad exp(ξˆi,1 θi,1 ) · · · exp(ξˆi, fi −1 θi, fi −1 ) ∈ R6×6 fi   i = Blockdiag ξi,1 , ξi,2 , · · · , ξi, fi ∈ R6 fi × fi Regarding the derivation of error models based on the local POE formula is a little bit abstract and complicated, the transformations of error components between different frames are explained in Fig. 2. First, in the local POE model, the twist coordinates pi, j are used to represent the kinematic parameters which includes the structural parameters of the corresponding link and the motion errors of the joints. But it should be noted that the motion errors in pi, j only stand for the joints’ zero-position errors, namely, the home errors of the actuated joints and position errors of passive joints caused by the encoder offsets and idle motions, respectively, at the manipulators home position. Thus, the 6 × 6 matrix Ai, j is introduced to map the parameter errors to the twist one of local frames. The twist error Ai, j δ pi, j is represented in the link’s local frame, so adjoint operator Ad(gi,0 · · · gi, j−1 ) is used to transform it to the spatial frame at home position. Then, the motion of joints is taken into account to obtain the links’ twist errors in {S} at the current configuration as Ad(exp(ξˆi,1 θi,1 ) · · · exp(ξˆi, j θi, j ))Ad(gi,0 · · · gi, j−1 ) Ai, j δ pi, j . Since all the links’ kinematic errors have been uniformly represented in twist forms with respect to the spatial frame, they can be directly summed up to achieve the pose error of the end-effector with respect to {S}, namely δ yi as presented in (7). On the other hand, the errors δθ i in (7) represents the joints motion errors which exclude the ones in pi, j . In other words, δθ i will be raised only after the manipulator deviating from its home position and they should be divided into passive and active situations according to their different properties. On one hand, for active joints, the motion errors only depend on the sensor noises, and these errors are usually much smaller that the kinematic errors. Therefore, the motion errors of the active joints can be ignored in the error modeling procedure. However, on the other hand, it is known that the passive joints in parallel manipulators perform idle motions to adjust the kinematic inconsistence of limbs due to geometric errors. So, their displacement errors do not keep invariant, but change in different configurations. Therefore, such kind of errors should be eliminated from the system error model of the whole parallel manipulator, which will be discussed in the following section. For an active joint, the motion error δθ i, j refers to the home error due to the misalignment of {Ri, j } with {Fi, j−1 } about the joint axis at the initial position. In the local POE formulation, these errors can be integrated into the limbs’ geometric ones by means of rotating the local frames about their z-axes as





gi, j exp ζˆi, j (θi, j + δθi, j ) = gi, j exp(ζˆi, j δθi, j ) exp(ζˆi, j θi, j ) = gi, j exp(ζˆi, j θi, j )

(8)

Please cite this article as: G. Chen et al., Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula, Mechanism and Machine Theory (2017), https://doi.org/10.1016/j.mechmachtheory.2017.11.003

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where gi, j = gi, j exp(ζˆi, j δθi, j ) = exp(pˆ i, j ) denotes the link’s geometry including the joint’s home position error. According to (8), it is known that both the geometric errors of links and the home position ones of joints have been integrated into the kinematic ones, namely δ pi , in the proposed error model (7). Therefore, only the identifiability of those parameters needs to be analyzed to eliminate redundant error components.

2.2. System error models of parallel manipulators As indicated in the above, the motion errors of passive joints can not be determined independently, such that they should be eliminated in advance to establish the system error model for the whole parallel manipulator. For ease of discussion, but without loss of generality, it is assumed that the first mi joints in limb i are passive ones. According to the reciprocal screw theory, for a non-singular configuration, there always exist 6 − mi independent twists which are reciprocal to all those passive joints [42–45]. Then, it yields T

ξ i,k i, j ξi, j ≡ 0, j = 1, · · · , mi , k = 1, · · · , 6 − mi

(9)

where ξ i,k ∈ R6×1 , k = 1, · · · , 6 − mi form a basis of the reciprocal screw system to the passive joints of limb. Here, =





03 I3 is the inverse operator satisfying −1 = . I3 03 The reciprocal relation (9) for all 6 − mi twists can be assembled in a matrix form as

Jr,i r,i r,i ≡ 0(6−mi )×mi

(10)

where the coefficient matrices can be constructed by the first mi blocks of i and i , which are associated with the passive joints, as



Jr,i = ξ i,1 , · · · , ξ i,6−m

T

i

∈ R(6−mi )×6

 

r,i = I6 , · · · , Ad exp(ξˆi,1 θi,1 ) · · · exp(ξˆi,mi −1 θi,mi −1 ) ∈ R6×6mi   r,i = Blockdiag ξi,1 , · · · , ξi,mi ∈ R6mi ×mi By projecting the pose error of the moving platform to the limb’s reciprocal system Jr,i , the reduced error model of the limbs can be rewritten as

Jr,i δ yi = Jr,i i i δ pi

(11)

where Jr,i r,i r,i ≡ 0. As a result, the kinematic error model of the whole parallel manipulator can be obtained by concatenating all those of the limbs together as

δ y = J−1 r Jr, p δ p = Jp δ p

(12)

where δ y = δ y1 = · · · = δ yn since the manipulator’s end-effector is shared by all limbs. While δ p = [δ pT1 , · · · , δ pTn ]T ∈

n

R(6n+6 i=1 fi )×1 represents the system error vector of the whole parallel manipulator. The coefficient matrices can be given by Jr = [JTr,1 , · · · , JTr,n ]T ∈ R6×6





Jr, p = Blockdiag JTr,1 1 1 , · · · , JTr,n n n ∈ R6×(6n+6

n

i=1 f i )

As indicated above, the parallel manipulators discussed in this paper are assumed neither over-constrained nor underactuated. Thus, the limbs’ reciprocal screw systems Jr,i , i = 1, · · · , n are linearly independent of each other and their union will span the whose space of se(3). Therefore, Jr is a 6 × 6 non-singular matrix and the system error model (12) can always be derived for the parallel manipulator at a non-singular configuration.

3. Identifiability analysis of the system’s kinematic parameters The system error model (12) includes all potential geometric errors of links and home position errors of active joints. Therefore, it is complete for the kinematic calibration of parallel manipulators. However, in such kind of error models, there are redundant components which will influence the computational stability and efficiency during the iterative identification of kinematic parameters. Hence, identifiability analysis should be performed to determine the independent components of the system error vector in the calibration model of the parallel manipulators. As a result, minimal error models can be obtained by means of eliminating the redundant components. Please cite this article as: G. Chen et al., Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula, Mechanism and Machine Theory (2017), https://doi.org/10.1016/j.mechmachtheory.2017.11.003

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3.1. Identifiability of the kinematic parameters in the limbs In our previous work [18], it has been proved, for a general serial robot, there exist 2r + 4 p redundant components in the error models based on the local POE formula, where r and p are the numbers of revolute and prismatic joints, respectively. Further, an analytical algorithm was developed to identify and eliminate these redundant components, such that error models satisfying the continuous, complete and minimal conditions can be achieved for the kinematic calibration. In this section, a similar method is used for the identifiability analysis of the kinematic parameters in the individual limbs of parallel manipulators. To make the derivation easy to understand, a general example is used to demonstrate the main idea of identifiability analysis in kinematic calibration. The calibration model of a robot can be represented in a general form as

y (θ ) = A (θ ) x

(13)

where x ∈ Rm×1 represents the vector of kinematic errors need to be identified. y (θ ) ∈ R f ×1 represents the pose error of the end-effector at the configuration θ . And A(θ ) ∈ R f ×m is the corresponding identification matrix. Then, the identifiability analysis of kinematic parameters is to determine an orthogonal matrix, whose blocks are associated with the null and row spaces of the identification matrix, respectively. Let QA = [NA , RA ] ∈ Rm× f be an orthogonal matrix, where the columns of NA ∈ Rm×n and RA ∈ Rm×r are associated with the null and row spaces of A(θ ). which yields QA QTA = Im×m , A(θ )NA = 0 f ×n and A(θ )RA RTA = A(θ ). Thus, the calibration model (13) can be reduced as

y (θ ) = A(θ )[NA , RA ] [NA , RA ]T x = A(θ )NA NTA x + A(θ )RA RTA x = A(θ ) x

(14)

∈ relates to the redundant error components because its coefficient matrix A(θ )NA vanishes. While x = comprises the independent error components can be identified and A(θ ) = A(θ ) RA ∈ R f ×r is the corresponding identification matrix. According to (14), it is known that the error components in the null space of identification matrix correspond to redundant error components which do not contribute to the pose error of end-effector at all. In other words, only those in the row space of A(θ ) can be identified via kinematic calibration. Therefore, the redundant components should be eliminated from the error model for the sake of computational stability and efficiency. However, the identification matrix A(θ ) is a function of the manipulator’s configuration. It means that the derived null and row spaces should be available for all (an infinite number of) configurations. And this is the main challenge in the identifiability analysis of robot kinematic calibration. In the rest of this section, based on the local POE formulation, an analytical approach is presented to determine the configuration-invariant solution to the null and row spaces of the identification matrix in the limbs’ error models. In limb error model (7), all configuration-related components have been stacked into the matrix i . Meanwhile, the block diagonal matrix i ∈ R6( fi +1 )×6( fi +1 ) is non-singular. Hence, the identifiability analysis can be transformed to the determination of the null space of i that is independent of limb configuration θ i . For simplicity, only the result is presented here and the detailed derivation can be found in [18]. Let Ui, j ∈ R6×ni, j be the matrix associated with the null space of Zξ = ad(ξˆi, j ) ∈ R6×6 , such that Zξ Ui, j ≡ 0. According where NTA x RTA x ∈ Rr×1

Rn×1

i, j

i, j

to the exponential map (35), it is easy to derive





Ad exp(ξˆi, j θi, j ) Ui, j = (I6 + bi j,1 Zξi, j + bi j,2 Z2ξi, j + bi j,3 Z3ξi, j + bi j,4 Z4ξi, j )Ui, j = Ui, j

(15)

where the coefficients bi j,k , k = 1, · · · , 4 are functions of the joint variable θ i, j as defined in (35). From (15), the columns of Ui, j are the eigenvectors of Ad(exp(ξˆi, j θi, j )) corresponding to 1. Regarding the block structure of i , the following identity can be obtained readily as

  (i, j+1 − i, j )Ui, j = Ad exp(ξˆi,1 θi,1 ) · · · exp(ξˆi, j−1 θi, j−1 ) Ad(exp(ξˆi, j θi, j )) − I6 Ui, j = 0, j = 1, · · · , fi

(16)

where i, j = Ad(exp(ξˆi,1 θi,1 ) · · · exp(ξˆi, j−1 θi, j−1 )) is the jth 6 × 6 block of i as presented in (7). By assembling the identity (16) for all j = 1, · · · , fi , the null space of i can be constructed in a matrix form as

⎡

⎢ ⎢ ⎢ i Ui ≡ 0 ⇒ Ui = ⎢ ⎢ ⎢ ⎣

where Ui ∈ R6( fi +1 )×

fi

−Ui,1

0

Ui,1

−Ui,2

0 .. . 0

Ui,2 .. . ···

n i=1 i, j

··· .. . .. . .. . 0

0 .. .

⎤

⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎦

(17)

−Ui, fi Ui, fi

corresponds to the null space of i .

Please cite this article as: G. Chen et al., Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula, Mechanism and Machine Theory (2017), https://doi.org/10.1016/j.mechmachtheory.2017.11.003

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Fig. 3. Determination of the null space for the adjoint twist representation of revolute and prismatic joints.

According to the definition, the rank of Zξ = ad(ξˆi, j ) is 4 and 2 for revolute and prismatic joints, respectively. Moreover, i, j the corresponding null space can be obtained as

  ⎧ ωˆ i, j ωi, j ⎪ ⎪ ξ = ⇒ Z = ⎪ ξi, j ⎨ i, j ⎪ ⎪ ⎪ ⎩ξi, j =



vi, j 03

ωi, j

vˆ i, j





⇒ Zξi, j =

03

ωˆ i, j

03



ωˆ i, j 03 03



⇒ Ui, j =

ωi, j

03

vi, j

ωi, j

 ⇒ Ui, j =

ωi, j

03

03

I3



, ni, j = 2 for revolute joints 6×2



(18) , ni, j = 4 for prismatic joints

6×4

where the components ωi, j and vi, j of joint twist ξ i, j are independent of limb configuration and can be specified in terms

of link kinematics, as shown in Fig. 3. It is known that the columns of Ui, j correspond to the commutative elements of ξ i, j in se(3) [41].

fi According to (18), the rank of Ui can be obtained as ni = j=1 ni, j = 2ri + 4 pi , where ri and pi are the numbers of equivalent revolute and prismatic joints in the limb, respectively, so ri + pi = fi . Then, a 6( fi + 1 ) × 6( fi + 1 ) orthogonal matrix can be constructed as Ni = [U⊥ i , Ui ] satisfying ⊥T Ni NTi = U⊥ + Ui UTi ≡ I6 i Ui

(19)

6( fi +1 )×(2ri +4 pi ) is the normalized matrix which is column equivalent to U , such that U⊥T U⊥ ≡ I where U⊥ 2ri +4 pi . While i i ∈R i i 6 ( f +1 ) × ( 4ri +2 pi +6 ) is the one orthogonal complement to U⊥ , which is associated with the row space of  . Thus, it Ui ∈ R i i i ⊥T T satisfies UTi U⊥ i = Ui Ui ≡ 0 and Ui Ui ≡ I4ri +2 pi +6 .

Analogous to (14), the error components in the null space of i can be directly eliminated using the orthogonal partitioning matrix Ni and the reduced error model yields

Jr,i δ yi = Jr,i i Ni NTi i δ pi = Jr,i i Ui UTi i δ pi

(20)

0. δ pi ∈ R(4ri +2 pi +6)×1 relates to the error components in the row space of i . And is the corresponding coefficient matrix. The result (20) is similar to the one derived for serial robot calibration [18]. The redundant components to the coefficient matrix i have been eliminated from the error model using the orthogonal partitioning matrix Ni . And the corresponding i Ui is full column rank. However, due to the existence of passive joints, the limb error models of parallel manipulators are slightly different from those of serial robots. Therefore, the identifiable parameters can be determined once the influence of passive joints, namely the factor Jr,i , on the column dependency of i Ui is revealed. According to (19), it is easy to derive i Ui UTi ≡ i . Then, based on the reciprocal relation (10), the null space of the reduced identification matrix Jr,i i Ui can be obtained as ⊥T where Jr,i i U⊥ i Ui i δ pi ≡ ( 6 −m ) × (2ri +4 pi +6 ) i Jr,i i Ui ∈ R

UTi i

Jr,i i Ui UTi i = Jr,i i i = Jr,i r,i i ≡ 0 [Tr,i ,

(21)

R6( fi +1 )×mi ,

where i = ∈ such that i i = r,i r,i since r, i only consists of the first mi blocks of i as defined in (10). (4ri +2 pi +6 )×mi and V ∈ From (21), two matrices corresponding to the null and row spaces of Jr,i i Ui , termed V⊥ i i ∈R T ( 4 r +2 p +6 ) × ( 4 r +2 p +6 −m ) i i i i , respectively, can be constructed from the item U  . As a result, another orthogonal partitioning R i i i matrix can be defined as Mi = [V⊥ i , Vi ] satisfying 0T ] T

⊥T Mi MTi = V⊥ + Vi VTi ≡ I 4ri +2 pi +6 i Vi

where

V⊥ i

and Vi are column equivalent and orthogonal complement to

(22) UTi

i , respectively.

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Using the orthogonal partitioning matrix (22), the redundant error components in the null space of Jr,i i Ui can be further eliminated from the reduced error model as

Jr,i δ yi = Jr,i i Ui Mi MTi UTi i δ pi = Jr,i i Ui Vi VTi UTi i δ pi = Jpi δ pi

(23)

where δ pi = VTi UTi i δ pi ∈ R(4ri +2 pi +6−mi )×1 relates to the independent error components of the limb’s kinematic parameters. Jpi = Jr,i i Ui Vi ∈ R(6−mi )×(4ri +2 pi +6−mi ) is the corresponding error transferring matrix. Relation (21) indicates that there exist mi extra redundant components in the limbs’ kinematic error vector due to the existence of passive joints. Using the orthogonal partitioning matrix Mi , these redundant components can be eliminated in a straightforward way, as indicated in (23). In summery, there are 4ri + 2 pi + 6 − mi independent error parameters can be identified for each limb of the parallel manipulator via the kinematic calibration. 3.2. Identifiability of the kinematic parameters in the parallel manipulators Analogous to (12), the system error model without redundant error components can be obtained by concatenating (23) for all limbs of the parallel manipulator as

δ y = J−1 r Jp δ p = Jp δ p where δ p = [δ pT1 , · · · , δ pTn ]T ∈ R(6n+

J−1 r Jp



(24)

n

i=1 (4ri +2 pi −mi )



where Jp = Blockdiag Jp1 , Jp2 , · · · , Jpn ∈ R

)×1 is the reduced error vector of the whole parallel manipulator. Jp =

6× ( 6n+

n

i=1 (4ri +2 pi −mi )

.

In the system error model (24), the components of the error vector δ p can be expressed as δ pi = VTi UTi i δ pi . Since the multiplier VTi UTi i Ai is full row rank, δ pi is an independent vector as same as the original δ pi , so is δ p. Hence, the identifiability of error parameters in (24) is simply equivalent to the linear independency of columns in the identification matrix Jp = J−1 r Jp . As mentioned above, both and Jr are invertible, so only Jp needs to be examined. Since Jp is a block diagonal matrix, any two columns belonging to different blocks are independent of each other. Meanwhile, the independency of columns in the same block has been verified in the above section. Thus, Jp is full column rank, too. Consequently, the conclusion can be made that the error parameters in the system model (24) of parallel manipulator are independent of each other. In other words, all components of δ p can be identified via kinematic calibration. Therefore, the number of identifiable error parameters in a parallel manipulator equals to the dimension of the error

vector δ p, namely 6n + ni=1 (4ri + 2 pi − mi ). As indicated in the above, the parallel manipulators discussed in this paper

are assumed neither over-constrained nor under-actuated. Then, it yields ni=1 (6 − mi ) = 6 ⇒ ni=1 mi = 6n − 6. As a result, the number of identifiable error parameters in the parallel manipulators can be represented in a concise form as

Nident =

n 

( 4ri + 2 pi + 6 − mi ) = 4r + 2 p + 6

(25)

i=1

where r = ni=1 ri and p = ni=1 pi denote the total numbers of equivalent revolute and prismatic joints in the parallel manipulator, respectively. The result (25) shows that the number of identifiable error parameters of a parallel manipulator is uniquely determined by the quantities of equivalent revolute and prismatic joints in it. It is independent of the manipulator’s concrete topology, such as the number of limbs, the quantities of the active and passive joints, the geometric constraints between the joints, and so on. Moreover, it exhibits a fascinating connection with the one derived from the serial robots. Hence, for both parallel and serial manipulators, the maximum numbers of identifiable parameters can be uniformly represented by 4r + 2 p + 6. 4. Algorithm for the identification of kinematic parameters in parallel manipulators Based on the identifiability analysis presented in the above section, a general strategy for the parameter identification in the kinematic calibration of parallel manipulators is proposed in this section. As illustrated in Fig. 4, the main procedures of the identification process are similar to those for serial robots. The kinematic parameters are iteratively updated according to the identification algorithm until the calculated poses of the manipulator’s end-effector are close enough to the corresponding measured ones. As it is known, it is really difficult to achieve the closed-form solution to the forward kinematics of parallel manipulators, so a numerical searching method is introduced to specify the actual pose of the parallel manipulators with inaccurate kinematic parameters. In such a phase, the ‘actual’ values of the manipulator’s kinematic parameters are replaced by the identified ones in current step. Meanwhile, the active joints are accurately actuated. Therefore, determination of the passive joints’ displacements becomes the key point of the searching algorithm since they are adjusted according to the kinematic errors. For any given pose of the parallel manipulator’s end-effector, the nominal inverse kinematics is first performed to determine the ideal joint variables in the limbs, which are regarded as an initial guess of the searching algorithm. In this phase, Please cite this article as: G. Chen et al., Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula, Mechanism and Machine Theory (2017), https://doi.org/10.1016/j.mechmachtheory.2017.11.003

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Fig. 4. Flowchart of the iterative identification process for the kinematic parameters of parallel manipulators.

only the motion errors of the passive joints are taken into account. According to the consistency of the end-effector, the constraint equation can be obtained for any two limbs as

gi,st = g j,st ⇒ g i,st + J θ θ i = g j,st + J θ θ j i j

(26)

where gi,st and g j,st represent the end-effector poses in limb i and j, respectively. While J θ and J θ are the corresponding i j Jacobian of the passive joints. Then, the system constraint equation of the whole parallel manipulator can be obtained by means of concatenating (26) for all limbs as

Jθ θ = y

(27)

T

T

where θ = [θ 1 , · · · , θ n ]T and y = [yT1,2 , · · · , yTn−1,n ]T . yi, j = (log(g j,st g−1 ))∨ represents the current pose differi,st ence between the limbs. And the system Jacobian matrix can be obtained as

⎡

Jθ

1

⎢ ⎢0 Jθ = ⎢ ⎢. ⎣ .. 0

−Jθ Jθ ..

2

2

. ···

···

0 −Jθ ..

. 0

..

.

..

.

⎥ ⎥ ⎥ ⎥ 0 ⎦

3

Jθ

n−1

⎤

0 .. .

−Jθ

n

The actual displacements of the manipulator’s passive joints which are influenced by the kinematic errors of the links, can be specified by solving (27) iteratively as

θ k+1 = θ k + θ k = θ k + (Jkθ )−1 yk

(28)

Please cite this article as: G. Chen et al., Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula, Mechanism and Machine Theory (2017), https://doi.org/10.1016/j.mechmachtheory.2017.11.003

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11

which is terminated when the forward kinematics of the limbs are in accordance with each other. Accordingly, the actual pose of the parallel manipulator’s end-effector can be directly determined according to the POE formula (3) of any limbs. Then, by measuring enough configurations for the parallel manipulator, the system calibration model can be obtained as

⎡

δ y1

⎡

⎤

⎤

Jp,1 ⎢ ⎥ δ Y˜ = ⎣ ... ⎦ = ⎣ ... ⎦δ p = ˜Jp δ p δ ym Jp,m

(29)

where δ yk , k = 1, · · · , m are the deviations of the end-effector’s calculated poses from the measured ones, given by

 ∨ δ yk = log(gst,k g−1 ) st,k

(30)

where gst,k denotes the measured pose of the end-effector at the configuration k. While gst,k is the calculated one with the kinematic parameters identified at the current step. T Since the identification matrix ˜Jp is full column rank, ˜Jp ˜Jp relates to a (4r + 2 p + 6 ) × (4r + 2 p + 6 ) non-singular matrix. Then, the least-square estimation for the independent kinematic errors δ p can be obtained as +

T

T

δ p = ˜Jp δ Y˜ = (˜Jp ˜Jp )−1 ˜Jp δ Y˜

(31)

+ T T where ˜Jp = (˜Jp ˜Jp )−1 ˜Jp is the pseudo-inverse of ˜Jp In the reduced error model, the redundant components have been eliminated, so the dimension of the independent error vector δ p is less than the number of the parallel manipulator’s kinematic parameters. Therefore, an extra update theme for the manipulator’s kinematic parameters is introduced by letting the redundant errors always be zero. Then, we have



δ pi =

VTi UTi

i δ pi ⇒ δ pi = 

−1 i

Ni M i

δ pi



(32)

0

where δ pi is the independent error components in the limb i, which can be extracted from the system one δ p as in (31). Mi and Ni are the orthogonal partitioning matrices which are updated according to the current kinematic parameters. Finally, an iterative algorithm can be used to minimize the linearization error of the calibration model as

pk+1 = pk + δ pk pk

(33)

δ pk

where represents the estimation of the kinematic parameter at the iteration k and is the corresponding kinematic error vector according to (31) and (32). ˜ converges to some stable value. The iterative algorithm will be repeated until δ p approaches zero and the residual δ Y Since the system identification matrix for the kinematic calibration is full column rank, computational stability and efficiency can be guaranteed for the parameter identification process. As a result, the resultant vector p∗ can be considered as the best estimation of the kinematic parameters in the calibration of parallel manipulator. 5. Conclusion In this paper, a systematic approach is proposed to establish the complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on the local POE formulation. The identifiability analysis proves that the maximum number of identifiable kinematic parameters in a non-overconstrained parallel manipulator is 4r + 2 p + 6, where r and p denote the numbers of the equivalent revolute and prismatic joints, respectively. This result is in accordance with the acknowledged conclusion for their serial counterparts. By introducing orthogonal partitioning matrices, an analytical algorithm was proposed to determine and eliminate the redundant error components due to the existence of passive joints. In addition, a corresponding strategy for the kinematic-parameter identification of parallel manipulators is also presented to improve the efficiency of the identification process. Acknowledgments This research was supported in part by the Natural Science Foundation of China under grant 51505279 and 51421092, and by the National Basic Research Program of China under grant 2014CB046600. Appendix A. Mathematical preliminaries In order to make the algebraic derivation of the proposed method easy to understand, some related concepts in Lie group/algebra are presented here. Details can be found in the monographs by [41,46]. Please cite this article as: G. Chen et al., Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula, Mechanism and Machine Theory (2017), https://doi.org/10.1016/j.mechmachtheory.2017.11.003

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The matrix exponential map from the Lie algebra se(3) to the special Euclidean group SE(3) is defined as



R exp : se(3 ) → SE (3 ) : ξˆ → g = exp(ξˆ) = I4 + a1 ξˆ + a2 ξˆ2 + a3 ξˆ3 = 0

p



1

(34)

where g ∈ SE (3 ) is the homogeneous transformation matrix of rigid motions. R ∈ SO(3 ) represents the rotation matrix and p ∈ R3 is the vector of translation. Here, I4 denotes the 4-order identify matrix. And the scalar coefficients are a1 = 1, a2 = 1 (1 − cos θ ), a3 = θ13 (θ − sin θ ) and θ = ω. ξˆ ∈ R4×4 is an element of se(3) in its standard representation as θ2



ξˆ =



ωˆ

v

0

0

ˆ ∈ so(3 ) is the 3 × 3 skew-symmetric matrix associated with ω = [ω1 , ω2 , ω3 ]T ∈ R3 . v ∈ R3 is an arbitrary vector. where ω Using adjoint representation, the matrix exponential map (34) can also be expressed as



Ad(g ) = exp(Zξ ) = I6 + b1 Zξ + b2 Zξ + b3 Zξ + b4 Zξ = 2

3

4

R

0

pˆ R

R



(35)

where Ad(g ) ∈ R6×6 denotes the adjoint representation of g ∈ SE (3 ). pˆ ∈ R3×3 represents the skew-symmetric matrix of p. The coefficients are given by b1 = 21θ (3 sin θ − θ cos θ ), b2 = 2θ1 2 (4 − 4 cos θ − θ sin θ ), b3 = 2θ1 3 (sin θ − θ cos θ ), b4 = 1 (2 − 2 cos θ − θ sin θ ). Here, Z ∈ R6×6 is the adjoint representation of ξˆ ∈ se(3 ), given by 2θ 4

 ωˆ Zξ = ad ξˆ =

0

ωˆ

vˆ



ξ

∈ R6×6

which can also be identified with the vector ξ = [ωT , vT ]T ∈ R6×1 , referred to as the twist coordinates of the element ξˆ ∈ se(3 ). Let {A} and {B} be two coordinate frames, and gab ∈ SE (3 ) be the relative pose of {B} with respect to {A}. Then, the change-of-coordinates of an element ξˆ ∈ se(3 ) between these two frames can be represented as

ξˆa = gab ξˆb g−1 ⇒ ab

⎧  ∨ ⎨ ξa = ξˆa = Ad(gab ) ξb ⎩

(36)

Zξa = ad(ξˆa ) = Ad(gab ) Zξb Ad(g−1 ) ab

where Ad(gab ) ∈ R6×6 is the adjoint representation of gab , as indicated in (35). Accordingly, the coordinate transformation of the exponential map (34) can be derived as

gab exp

   ˆ g−1 = exp ξˆa ξˆb g−1 = exp g ξ b ab ab ab

(37)

According to the definition, the derivative of the exponential map (34) with respect to the element of se(3) can be derived as

  ∨  ∞ δ exp(ξˆ) exp(−ξˆ) = k=0

  1 Zkξ δξ = I6 + c1 Zξ + c2 Z2ξ + c3 Z3ξ + c4 Z4ξ δξ = Aξ δξ ( k + 1 )!

(38)

where Aξ = I6 + c1 Zξ + c2 Z2ξ + c3 Z3ξ + c4 Z4ξ ∈ R6×6 . And the coefficients are given by c1 = 2θ1 2 (4 − θ sin θ − 4 cos θ ), c2 = 1 (4θ − 5 sin θ + θ cos θ ), c3 = 2θ1 4 (2 − θ sin θ − 2 cos θ ) and c4 = 2θ1 5 (2θ − 3 sin θ + θ cos θ ). 2θ 3 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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Please cite this article as: G. Chen et al., Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula, Mechanism and Machine Theory (2017), https://doi.org/10.1016/j.mechmachtheory.2017.11.003