1 Introduction
The inverse kinematics (IK) problem plays an important role in robotics, computer games, graphics, and vision, as it is a fundamental building block for animating, controlling, tracking and reconstructing articulated objects, such as robotic arms or human bodies. The IK problem refers to the task of recovering parameters of a kinematic skeleton (e.g. joint angles), given a set of observed locations for (some of) the joints.
The kinematic skeleton is represented as a tree, where each node is a joint that has translational and/or rotational degreesoffreedom. In forward kinematics, one computes the positions of the joints given the translation and rotation parameters. Here, the transformation for each joint is defined relative to its parent joint, so that a chain of transformations is applied along the path from the root node to a leaf node (endeffector). Inverting this process, i.e. given some of the joint positions, one wants to recover the kinematic parameters that lead to this joint position configuration.
The IK problem is considered to be difficult due to several reasons. On the one hand, depending on the kinematic skeleton and the observed joint positions, the inverse kinematics can be illposed so that there may be multiple configurations of kinematic parameters that lead to the same joint positions. On the other hand, depending on the given joint locations, a solution that produces an exact fit to the observations might not exist. Lastly, the resulting optimisation problem is nonconvex, which makes it generally difficult to find a globally optimal solution. Hence, approximations are oftentimes used in practice, which in turn require an initialisation that is sufficiently close to a global optimum. In computer vision, one of the most dominant applications of the IK problem is tracking and reconstructing articulated objects based on a temporal sequence of data (e.g. depth images, or RGB images). A common approach for tackling tracking applications is to initialise the kinematic parameters for the next frame using the tracked result from the current frame, and then solve the IK problem using local optimisation methods. However, this is not possible for the very first frame of a sequence. Hence, in many works the authors assume that a good initialisation is available for the first frame, e.g. based on an initial calibration with a neutral pose, as done in
[33]. In contrast, our proposed method is entirely initialisationfree, and is therefore wellsuited for handling such cases. The main contribution of this work is a polynomialtime solution of the IK problem, coined SDPIK, that finds a global optimum of a (convex relaxation of) the IK problem based on semidefinite programming.2 Related Work
In this section we first address local optimisation methods for the inverse kinematics problem, followed by global methods. Subsequently, we summarise the most relevant works that consider semidefinite programming relaxations of related optimisation problems.
Local IK approaches:
Local optimisation methods seek to iteratively find a solution of the IK problem based on a given initial estimate. One class of such methods use the firstorder Taylor approximation of the problem and attempt to solve a linear system characterised by the Jacobian matrix. Alternatives are the MoorePenrose pseudoinverse method, the Jacobian transpose method (equivalent to gradient descent for leastsquares error), the LevenbergMarquardt method (equivalent to gradient descent for damped leastsquares error), and other variants
[6, 10, 14, 20]. Secondorder methods also exist, but such approaches require the computation of the Hessian of the forward kinematics function, which incur higher computational cost. QuasiNetwon methods, such as the BroydenFletcherGoldfarbShanno (BFGS) algorithm, have been used to provide faster approximations for solving the IK problem [23].Instead of trying to approximate and invert the forward kinematics function, heuristic methods employ simple rules to be iteratively followed and can often reach the IK solution. Cyclic coordinate descent (CCD) and its variants are heuristic methods that seek to minimise joint errors by changing one kinematic parameter at a time
[27, 38]. Forward and backward reaching IK (FABRIK) [4] provides a method that can provably converge to the correct solution, when feasible, for a single unconstrained kinematic chain [3]. Evolutionary algorithms such as particle swarm optimisation and genetic algorithms, are heuristics inspired by evolution and are used to solve the IK problem in
[34, 35].Due to their speed and easeofuse, local optimisation methods are widely used in applications such as character animation [20], motion retargeting [19], modelbased tracking [33, 36]
, postprocessing on pose estimation
[26, 28], and data visualisation [27]. However, despite their popularity, local IK approaches require a good initialisation for complex, realworld skeletons, as otherwise they are prone to converge to unwanted local optima.Global IK methods:
Global IK methods aim to avoid this tendency getting stuck in local optima, and instead seek to obtain a global solution. One way of of achieving initialisation independence is based on training a machine learning model to solve the IK problem. In
[11], the authors learn an inverse kinematics function which maps from joint locations to kinematic parameters using a structured learning method.An alternative to learning approaches are global optimisation methods. In [16], the authors tackle the problem of forceclosure grasp synthesis based on sequential semidefinite programming, where rotation constraints are modelled in terms of bilinear matrix inequalities involving quaternions and rotation matrices. An approach for a feasibility formulation of the IK problem based on mixedinteger programming (MIP) is presented in [15]
. The main idea here is to discretise all nonconvex constraints based on binary variables. The resulting problem is then solved with a branch and bound algorithm, which is known to have exponential worstcase time complexity.
Contrary to the discussed works, which address feasibility versions of the IK problem, and/or do not admit polynomialtime algorithms, we propose a principled polynomialtime approach for the IK problem. Moreover, we consider a leastsquares version of the IK problem, which is most relevant for the majority of applications in vision and graphics, such as for the tracking or the reconstruction of articulated objects.
Semidefinite programming relaxations:
Semidefinite programming (SDP) relaxations are a popular way for tackling nonconvex optimisation problems. Such methods have been successfully used for a range of different problems in vision and beyond, e.g. for graph matching [32, 40] or multigraph matching [7, 21], the rigid registration of pointclouds [22, 25], the segmentation of images [39], or for permutation synchronisation [13]. However, generally such approaches are computationally expensive, since many of the SDP relaxations are based on a lifting of the variables, so that the size of the optimisation problem increases quadratically when moving from the original nonconvex problem to the convex relaxation [21, 32, 40].
One scenario where SDP relaxations particularly stand out is in problems involving 3D rotation matrices. On the one hand, lifting a matrix variable of size merely results in a relatively small variable of size , so that such problems can be solved efficiently. On the other hand, some relaxations that involve a single rotation matrix have been observed to be tight in practice, i.e. even when solving a relaxation of a nonconvex problem, the sofound solution is a global minimiser of the original nonconvex problem. This has for example been empirically demonstrated in [9] for the registration of 3D objects (with known correspondence). Other approaches that consider SDP approaches for problems involving rotations have been demonstrated for SLAM [30], posegraph optimisation [12], or rotation averaging [5, 17, 37].
In our work we consider an SDP relaxation for the inverse kinematics problem, which involves a composition of several rotations that are propagated through the kinematic chain.
2.1 Notation
Here, we briefly outline the used notation. By we denote the identity matrix, by
we denote the vector of all zeros, and the operator
vectorises a given input matrix by concatenating all the columns of . For an integer we use the notation . For a matrix we write to denote the vector that is formed by the th column of , and analogously to denote the row vector that is formed by the th row of . Moreover, for a 3D vector , or a matrix , we use and to denote their respective representation in homogeneous coordinates. For being a matrix, the notation means that is symmetric positive semidefinite.3 Inverse Kinematics
In this section we describe our approach for tackling the inverse kinematics problem. To this end, we first define the forward kinematics model, which is followed by the precise statement of the problem.
3.1 Forward Kinematics Model
We assume that we are given a tree that defines the kinematic skeleton, see Fig. 2. The root of the tree has 6 degreesoffreedom (DOF), of which 3 account for the global rotation, and 3 account for the global translation. Each nonroot joint has between 0 and 3 rotational DOF, where each (nonzero) DOF accounts for a rotation of a given angle around a given axis. Joints without DOF are used to model the endeffectors, e.g. the fingertips in the human hand, cf. Fig. 2.
OneDOF representation:
W.l.o.g., for convenience, we redefine this generic kinematic skeleton in such a way that each (nonendeffector) joint only has a single DOF that allows for a rotation around a given axis. To this end, for each joint of the original skeleton that has more than 1 DOF, we simply introduce one or two additional auxiliary joints (placed at the same position) that account for the additional DOF. We emphasise that while this redefinition of the kinematic skeleton does not change its kinematic behaviour, such a representation is more convenient for defining our SDP relaxation, as we will become apparent in Sec. 4.2.
Kinematic skeleton:
Let be the total number of joints, where each joint now has at most 1 DOF (due to the redefinition of the skeleton). The global translation of the root is denoted by . For each subsequent joint transformation, let for denote the (unitlength) rotation axis of the th joint, and let be the “bonevector” of the th joint, i.e. the offset of joint in the coordinate system of its parent (cf. Fig. 2). For being the parameter of the th joint, by we denote the transformation from the coordinate system of joint to the coordinate system of its parent, represented in homogeneous coordinates. To be more specific, we have
(1) 
where the rotation matrix is obtained by Rodrigues’ rotation formula as
(2) 
Here,
is the skewsymmetric operator that generates a
matrix from a 3D vector (i.e. for we have ). For joints with DOF (i.e. for endeffectors), we define to be the identity matrix .Forward model:
For being the parameter vector that stacks all joint parameters, the forward model for computing the position of the th joint is given by
(3) 
where by we denote the path from the th joint to the th joint (from children to root) in the kinematic skeleton, and is the zero vector represented in homogeneous coordinates. For brevity, we will refer to all elements of as angles, even if they represent translations.
Joint angle constraints:
In addition, for each joint there is an interval that defines the range of valid values for so that it must hold that . For notational convenience, we define and write .
3.2 Problem Statement
We are interested in the problem of finding the parameters such that the forward kinematics model best explains a given set of 3D joint position observations. Let denote the subset of joints for which the 3D positions , are known. The IK problem can now be phrased as a (constrained) nonlinear leastsquares problem that reads
(4)  
Depending on the set , there may be multiple parameter vectors that all lead to the same configuration of joint positions for all . As such, for general IK problems of the form (4), the solution may not be unique since there can be multiple global optima. Most commonly, such problems are solved based on local optimisation methods, e.g. by modelling the hard constraints as penalty in the objective function and then using a gradient descent procedure for locally optimising the objective function. A major downside of using such iterative approaches is that one requires a good initialisation for , so that the optimisation does not result in an unwanted local optimum. We will tackle the problem of finding a good initialisation for Problem (4) based on a convex relaxation, as we will describe next.
4 Convex Relaxation for Inverse Kinematics
In order to achieve a convex relaxation of the IK problem, we will first redefine the problem as a nonconvex quadratically constrained quadratic programme (QCQP) [1]. Subsequently, we will introduce our convex relaxation based on semidefinite programming.
4.1 Inverse Kinematics as QCQP
Rather than phrasing the IK problem in terms of the parameter vector , we will directly optimise for rotation matrices and the global translation.
Global and relative rotations:
Let denote the global rotation of the th joint (i.e. relative to the root), and let denote the rotation of the th joint relative to its immediate parent, where we use the notation to indicate the parent of joint . For all joints we have the relation
(5) 
where we define .
Forward kinematics:
The position of the th (nonroot) joint is defined recursively as
(6) 
where is the 3D position of its parent, and we define .
Joint angle constraints:
For the th joint the rotation relative to its parent is constrained to be within the interval , see Sec. 3.1. In our reformulation we impose a similar constraint directly on the rotation matrix. In order to do so, we express in terms of a canonical rotation , where in our case we choose (w.l.o.g.) a rotation around the xaxis, so that we have the general structure
(7) 
As such, we can write , for being a suitably chosen matrix that is determined a priori (i.e. before optimisation). In order to impose the joint angle limits we enforce that and lie within the unit circle, i.e. we impose the convex constraint
(8) 
In addition, for , we consider the line passing through and , and enforce that the elements and of are within a halfspace defined by this line. This results in a linear inequality constraint in and . We use to refer to both of these (convex) joint angle constraints.
Rotation constraints:
The set of (proper) rotations can be defined with quadratic constraints as
where the crossproduct is used to implement the righthand rule in order to ensure that the determinant is .
QcqpIk:
With the above elaborations we can now formulate the IK problem as the QCQP
(9)  
s.t.  
where the constraints are applied for all .
4.2 Inverse Kinematics as SDP
Before we introduce our semidefinite programming relaxation of the IK problem, we briefly summarise the main idea of semidefinite programming relaxations for general QCQPs.
4.2.1 Semidefinite Relaxations of Generic QCQPs
A generic QCQP can be written in canonical form as
(10)  
s.t. 
where are given symmetric matrices (that are possibly indefinite). Note that by using a homogeneous coordinate representation, this form also allows for linear terms in the objective as well as for linear constraints. Commonly, such nonconvex QCQPs are solved by means of lifting, where an additional lifted variable of size is introduced. Based on the property that for a given matrix it holds that , we can rewrite Problem (10) as
(11)  
s.t. 
It is wellknown that the constraint is equivalent to
Since the left part is a convex cone constraint, to obtain a convex relaxation the rank constraint (that accounts for the nonconvexity) is dropped, which leads to the semidefinite programming problem
(12)  
s.t.  
4.2.2 Semidefinite Relaxation for IK
In order to obtain our semidefinite programming relaxation for the inverse kinematics problem, we systematically apply the elaborations in Sec. 4.2.1 to Problem (9). In the following we will elaborate on this.
Matrix multiplication constraints:
For three orthogonal matrices of size , the matrix constraint can equivalently be written as
(13) 
Similarly as above, by dropping rank constraints, we obtain a convex relaxation of this constraint as
Rotation constraints:
As indicated in Sec. 4.1, the constraint can be represented with quadratic equality constraints. A semidefinite relaxation of the constraint is achieved by working with the vectorised , i.e. , and introducing a lifted variable of size , as done in Sec. 4.2.1. The interested reader is referred to [9, 31], where further details about the lifting and the constraints can be found. We use the notation , where is a convex set, to indicate that is a pair of variables that satisfy the lifted rotation constraints.
Our convex relaxation:
We now state the convex relaxation of the inverse kinematics problem, which reads
(14)  
s.t.  
Practical considerations:
We solve our convex relaxation using the general purpose modelling tool Yalmip [24] that interfaces the MOSEK solver [2]. In order to get a tighter convex relaxation it helps to introduce convex relaxations of redundant constraints, e.g. for orthogonality one would use convex relaxations for both the constraints and , see [9]. Once we have found a solution to Problem (14), we first project the obtained matrices onto the set
by means of Singular Value Decomposition, and subsequently extract the joint angles
directly from the projected matrices. The soobtained joint angles are not necessarily an optimum of the original IK objective (4). Hence we use the found as initialisation for a trust region method as implemented in Manopt [8]. We refer to our overall approach as SDPIK.5 Experiments
In this section we experimentally demonstrate the benefits of our proposed approach based on two realworld kinematic skeletons. To this end, in Sec. 5.1 we first compare different local optimisation methods. Subsequently, in Secs. 5.2 and 5.3, we compare the bestperforming local optimisation method to our convex relaxation approach, where the effectiveness of the proposed SDPIK will become apparent. Since local IK methods are sensitive to initialisation, for every pose we uniformly sample multiple random initialisations (5 for the experiments in Sec. 5.1, and 20 for the experiments in Sec. 5.2 and 5.3) within the joint limit constraints, and then run local optimisation for each of the sampled initialisations. Note that we show the results obtained by all random initialisations in the comparisons. We measure the quality of the results based on the squareroot of the normalised in (4), which we denote as
(15) 
In our experiments, we use two realworld skeletons (hand and human body, see Fig. 2) in 100 different natural poses sampled from captured motion data sequences [18, 29]. The motion data is represented as a sequence of angle vectors that animate the hand and body kinematic skeletons with predefined bone vectors . We obtain the “observed joint positions” for each pose by computing the forward kinematics using the angle vector . For the evaluation we consider two different cases:

The observed joint positions are noisefree, i.e. there exist kinematic parameters that yield an exact fit with an objective value of (cf. Sec. 5.2).

The observed joint positions have added noise, i.e. the existence of kinematic parameters for an exact fit is not guaranteed (cf. Sec. 5.3). This is the common case for practical applications.
5.1 Local Optimisation Methods
To choose a representative local optimisation method as baseline, we compare four methods on the task of fitting the hand skeleton to observed joint locations when all joints are observed (noisefree). For tackling the IK problem with local optimisation methods it is common practice to convert it to an unconstrained optimisation problem, where the joint angle constraints are modelled as penalties. As such, we obtain the differentiable optimisation problem
(16) 
where measures the distance of from the interval of plausible angles (as done in [28]), and
is the hyperparameter that trades off joint limit violations with joint position errors. For our experiments we fix
.In Fig. 3 we show results of gradient descent (GD), the trust regions method (TR), conjugate gradient (CG), and limited memory BroydenFletcherGoldfarbShanno (LBFGS) algorithms as implemented in the Manopt Matlab toolbox [8]. For all methods we used default parameters and allowed a maximum time budget of s. Each method is tested with the same 8 poses with 5 random initialisations, so that a total of IK problems is solved. While the first three methods yield solutions with similar quality, LBFGS performs significantly worse. Additionally, gradient descent and conjugate gradient are significantly slower compared to the other two methods. Hence we decide to use the trust region method (TR) as representative local IK method for the remaining experiments due to its good tradeoff between speed and accuracy.
5.2 Fitting to NoiseFree Observations
In Fig. 4 we show quantitative results when solving the IK problem for noisefree observations. Our SDPIK method almost always achieves an exact fit, i.e. the objective function value reaches 0, and thereby consistently outperforms the local optimisation approach. This occurs both when all joints are observed (All), and for the more ambiguous case where only the endeffectors and root are observed (End+Spine or End+Wrist), also cf. Fig. 2
. Note that the local IK method suffers more from outliers due to bad initialisations, while our method can consistently find better minima.
5.3 Fitting to Noisy Observations
In many situations a given inverse kinematics problem can potentially be infeasible, i.e. kinematic parameters that exactly explain given observed joint locations may not exist. In practice, this is the case when the joint position observations are noisy. For example, neural networks have been employed successfully for joint location prediction for hands or human bodies. However, these networks mostly output a set of independent joint locations without any constraints to comply with the underlying kinematic structure. Hence, some approaches subsequently fit a kinematic skeleton to the predictions to optimise for plausible joint angles, usually relying on local IK methods like gradient descent. The aim then is to find kinematic parameters
and hence a kinematically consistent and plausible set of joint positions that achieve a minimum distance to the observed locations, as e.g. done in [26, 28].In this experiment we mimic such a setting by fitting a kinematic skeleton to joint locations that exhibit noise, so that an exact fit may not be possible. To this end, we use the same poses as for the noisefree fitting in Sec. 5.2, and add noise to all observed joint locations. For each joint, the 3D noise offset is obtained by uniformly sampling a unit length direction , and a magnitude from , where is mm for the hand skeleton, and mm for the body skeleton.
For infeasible cases, the minimum value of the IK cost function in (15) fluctuates randomly with the sampled noise. The resulting box plot would merely capture how much each noisy sample violates the kinematic constraints instead of the quality of the solution. Hence, instead of reporting the raw value of the IK cost, we provide normalised cost values. To this end, for each pose we subtract the value of the smallest cost that is obtained by local optimisation among all the random initialisations.
In Fig. 5 we show that our method consistently achieves similar, or better performance than a local optimisation for the IK problem. Qualitative results for the local optimisation method, the projected solution of the semidefinite programming approach in (14), and our SDPIK are shown in Fig. 6.
6 Discussion & Limitations
Currently, solving the semidefinite programming problem (14) takes about s on average on an Intel i5 CPU with GB of RAM, both for the hand as well as the human body skeleton. We emphasise that our current implementation is rather prototypical and is written so that it can work with generic kinematic skeletons. Moreover, we did not conduct any optimisations to improve the computational efficiency—we expect that runtime improvements by one order of magnitude or even more could be achieved using a problemspecific efficient implementation.
Currently, our model does not support translational DOF in nonroot joints. In the future, we plan to also look into this case, as translational DOF can account for small changes in the bone length, which in turn can deal with the variance of bone lengths in realworld data.
7 Conclusion
We have presented a convex optimisation approach for the inverse kinematics problem based on a semidefinite programming relaxation. A major benefit of this approach is that we can find the global optimum of a (relaxation of the) IK problem, in contrast to local optimisation methods that heavily rely on good initialisations. Our experiments confirm this advantage and also demonstrate that our proposed SDPIK approach is a useful method for tackling inverse kinematics problems as they appear in computer vision and computer graphic problems.
Acknowledgements
This work was funded by the ERC Consolidator Grant 4DRepLy. We thank Marc Habermann for making the human body skeleton and motion sequences available to us.
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