Method of command control for a robot manipulator

[0002] The present invention relates to a method of command control for a manipulator, based on end-effector destination shifts (Δx_d) commanded by a programmer or a superordinate task with the aid of a manual control ball or the like, in combination with a calculation of articulation position values according to an algorithm of inverse kinematics.

[0003] Methods of command control of a manipulator based on end-effector destination shifts, in combination with a calculation according to an algorithm of inverse kinematics using the Jacobi Matrix, are known from Siciliano, B., Sciavicco, L.: “Modeling and Control of Robot Manipulators,” McGraw-Hill Companies (1996), pp 95 to 101, and from Vukobratovic, M., Kircanski, N.: “Kinematics and Trajectory Synthesis of Manipulation Robots,” Springer-Verlag, Tokyo, 1986, pp 105 to 122.

[0004] In this context, a method of inverse kinematics is obtained for kinematically non-redundant manipulators, albeit in a partial space of the working area that contains non-singular positions. Since singular configurations, i.e., articulator positions in which the Jacobi Matrix experiences a reduction in priority, do occur in practice, this method has only limited application, because the operation being performed with the manipulator (interactive path guidance, force control) must be aborted or delayed when a singular configuration (singularity) is approached.

[0005] In this context, a method is known that includes a calculation of a generalized inverse of the Jacobi Matrix. This method possesses a few advantageous properties, but is also associated with a number of disadvantages. For example, a smoothness of the calculated articulation path and low wear of the robot drive are attained through minimization of the local articulation position offset Δq (local energy criteria). In addition, the spacing between the articulation positions and the physical articulation stops is taken into consideration through the optimization of global criteria in the zero space of the Jacobi Matrix. Disadvantages of this method are that it cannot guarantee that path limitations will be maintained with physical articulation stops, and unstable behavior occurs in singular robot positions due to a (generalized) inversion of the Jacobi Matrix, and inefficient robot path courses are possible when conflicting local and global criteria exist.

[0006] In another known method, in which a calculation of the transposed Jacobi Matrix is performed, the commanded end-effector destination is attained iteratively corresponding to a representation of reverse kinematics as an optimization problem. An advantageous feature of this method is a stable behavior in singular robot positions, because the Jacobi Matrix is not inverted.

[0007] However, there is no guarantee that path limitations can be maintained through physical articulation stops and maximum articulation speeds. Superordinate heuristics must be constructed to meet these requirements, which then results in errors in the real end effector position compared to the desired end effector position. Inefficient path courses result in the form of interference movements of the end effectors, because the Cartesian linear movement commanded by the manual control ball cannot be transferred exactly to the end effector of the manipulator.

[0008] Moreover, the robot drive is subjected to high material wear due to abrupt passage through singular robot positions and because of a generally insufficiently smooth articulation path, since neither the weighted local articulation position offset nor the local articulation speed offset (local energy and acceleration criteria) have been optimized. A further negative consequence is a low convergence speed, i.e., a reduced ability to operate in real time, because no practical optimum strategy is known for determining the positively-defined Cartesian stiffness matrix.

[0009] DE 33 44 633 C2 describes a real-time control in which the redundant articulations that are not necessary for the movement of an end effector are noted for calculating the articulation speed, which simplifies the calculation of the inverse Jacobi Matrix. This type of calculation is performed for at least one of the articulation combinations. The speeds for each articulation are then determined through averaging of the calculated articulation speeds. A weighting of the articulation speeds thus takes place in this prior art.

[0010] U.S. Pat. No. 5,430,643 also describes a real-time method, in which the inverse Jacobi Matrix is calculated. The method known from this U.S. patent document also takes into consideration weight values for the articulation speeds, as well as path limitations for at least a graphic simulation of the robot's movements.

[0011] The German patent application 197 03 915.4 and U.S. application Ser. No. 08/017,485 propose a method in which an interactive path guidance of a kinematically redundant manipulator can be performed efficiently, with the advantage of an accordingly less complex and thus more user-friendly parametrization. The only disadvantage of this method is that it is not suitable for force control purposes since no uniform measurement exists to indicate which percentage of the desired end-effector destination shift can be attained; also, the method places a lower priority on attaining the best possible end-effector destination shift.

[0012] It is an object of the invention to provide a method of inverse kinematics command control for an interactive path guidance, and/or as a modular component of a superordinate task (e.g., force control) of a manipulator with an optimized acceleration behavior, with which path limitations can be reliably maintained through physical articulation stops and maximum allowable articulation speeds to ensure the accuracy of the determined solution with respect to the allowable path limitations, and with which the stress on the manipulator drive device is kept to a minimum by optimizing the acceleration behavior of the articulation axes.

[0013] In accordance with the invention, which relates to a method of command control for a manipulator of the above type, this object is accomplished in that

[0014] a new articulation position (q_i+1) of the manipulator is calculated, beginning with a commanded end-effector destination shift and the current actual value (q_i) of the articulation position of the manipulator,

[0015] with consideration of a quality function (f(q) ) to be minimized, which is parametrized by non-negative weighting values (aα_j, β_j, γ_j),

[0016] and with consideration of path limitations through physical articulation stops (q_min, q_max), maximum articulation speed ({dot over (q)}max), maximum articulation acceleration ({umlaut over (q)}max) in an environment of physical articulation stops, and the kinematic equation, which is represented by the Jacobi Matrix (J (q)), which articulation position predetermines the new values for the articulation regulators, with the quality function (f(q)) being the sum of energy criterion, reference-position criterion, acceleration criterion and an additional criterion,

[0017] the energy criterion being calculated from

[0018] (q−q_i)^tdiag(a_j)(q−q_i)

[0019] the reference-position criterion being calculated from

[0020] (q−q_ref)^tdiag(β_jj)(q−q_ref),

[0021] the value q_refbeing a predetermined articulation position value that is determined such that the sequence of calculated articulation position values (q_i) runs near this reference position value;

[0022] the acceleration criterion being calculated from

[0023] (q−2q_i−1)^tdiag(γ_j)(q−2q_i+q_i−1),

[0024] and

[0025] the additional criterion from −p, with the scalar parameterp satisfying the kinematic equation p·Δx_d=J(q_i(q−q_i) and the inequality 0≦p≦1; p·100 being the attained percentage of the commanded end-effector destination shift (Δx_d),

[0026] beginning with an articulation-position value q_ias the starting point, an allowable optimization vector is determined on the basis of the quality function with respect to all active secondary conditions that indicate which path limitations have been attained, and this vector is scaled according to the inactive secondary conditions that indicate which path limitations have not been attained;

[0027] the scaled optimization vector is added to the articulation position value calculated in the previous iteration step;

[0028] and the optimum quality of these articulation-position values is evaluated based on the quality function and the secondary conditions activated in the newly-calculated articulation position.

[0029] The method according to the invention may be used in the interactive path guidance of the manipulator based on end-effector destination shifts commanded by an operator with the aid of a manual control ball (space mouse) or the like.

[0030] The method according to the invention may, however, also be used as a modular component of a superordinate task, with end-effector destination shifts commanded by the superordinate task. A superordinate task may, for instance, be one of the type posed by force-control objectives.

[0031] The method of the invention results in low wear of the manipulator drive through minimization of the local articulation-position offset Δq and the local articulation-speed offset (energy criteria and acceleration criteria). “Narrow” articulation paths around a reference position (e.g., zero position) are attained, resulting in a reversible behavior of the manipulator movement.

[0032] This specifically serves to prevent the manipulator roller axes from twisting increasingly during the performance of the commanded manipulation until they come to rest against the end stop, rendering a continued performance of the manipulation task impossible.

[0033] Articulation stops are advantageously mostly avoided or approached gently or with low wear. This behavior is caused by a delay effect of the robot axes, which acts proportionally to the distance from the reference position, and by explicit acceleration limitations. In addition, a stable behavior results for singular robot positions, because the Jacobi Matrix is not inverted.

[0034] Another advantage is the occurrence of efficient path courses through the exact transfer of the Cartesian linear movement commanded by the manual control ball to the end effector of the manipulator. In addition, path limitations are reliably maintained through physical articulation stops and maximum articulation speeds.

[0035] Since this results in a restriction of the number of solutions, i.e., in a limited number of allowable articulation positions, a high convergence speed is attained with a gain in real-time operability. In addition, the attained percentage of the desired end-effector destination shift is calculated. Singularities can be approached in a stable manner and/or passed through; as a result it also is not necessary to limit the working space.

[0036] A Cartesian end-effector destination shift Δ_c, commanded at a time T_ican be commanded by the operator and/or by a superordinate task (e.g., force control) in the form of a 6-dimensional increment vector. Alternately, the increment vector may also be determined by evaluating the difference between absolute end-effector coordinates.

[0037] In the appended drawings:

[0038] FIG. 1 is a schematicized illustration of an input/output flux of reverse kinematics in the real time grid;

[0039] FIG. 2 shows, in the form of a block diagram, an algorithm of inverse kinematics used in the method of command control of a manipulator according to the invention, and

[0040] FIG. 3 shows the application of the method of the invention for the command control of a manipulator within the frame of the performance of a superordinate task, such as a force control.

[0041] The present invention is a method for transforming the input commands to a robot (or “manipulator”), preferably via Jacobian matrix methods. The invention also contemplates an apparatus for transforming the input commands, which may be interposed between an actuator and the robot. The actuator may any sort, e.g. a mouse or joystick, a recorder with a recorded sequence of commands to be output to the calculating unit, a second programmed computer, and so on.

[0042] The apparatus, which acts as a calculating unit, in most cases will be a microprocessor, computer, etc., but may be an analogue computer or other device which can carry out the method of the invention. The present invention also includes a combination of actuator, calculating unit, and robot.

[0043] FIG. 1 is a time graph showing the notation used for the inputs and outputs to the calculating unit of the present invention. The horizontal axis represents time. Calculation inputs are generically denoted x and results or outputs corresponding to robotic position are generically denoted q, with superscripts showing the corresponding instant of time. FIG. 1 shows that the calculations are preferably not continuous but incremental. The method calculates over the time interval a next qⁱ⁺¹from the preceding qⁱ.

[0044] A robot's manipulator will ordinarily have six degrees of freedom in space (three of translation and three of rotation). According to the particular geometry of the robot's limbs, rotators, etc., there will be several different sets of actuators and sensors associated with respective dimensions. The notation of FIG. 1 represents only one of several dimension, i.e. angles or directions, in which the manipulator can move.

[0045] It is to be noted that motion in any one of the six directions is likely to cause motion in at least one of the others, and these motions are interrelated according to the geometry of the robot. For example, an angular changes at a “elbow” will cause a “hand” to translate through an arc. If the input device or instruction set has a different geometry from the robot—for example, the input device has three levers for three directions in space but the robot has a single extendable arm—then the end-effect destination shift commands will need to be converted into manipulator articulation positions by the use of matrix methods. Thus matrix methods must be used in the present equation.

[0046] FIG. 2 is a flowchart schematically illustrating the method of the invention. Aside from the input from the actuator, i.e. Δx_d, the calculation process also takes into account parameters shown in the box at left.

[0047] One of these parameters is a limit on the input, Δx_max, which cuts off large commands, i.e. in case of a sudden joystick lurch or the like.

[0048] Other parameters limit the motion of the robot: displacement or angle limits are denoted by q_minand q_max, and maximum robot change rates and acceleration q_maxand q_max. In FIG. 3 these are shown as “Path limitations”.

[0049] Still other parameters α, β, and γ are described as “algorithm parameter weightings” in FIG. 3. These weighting values are preferably non-negative, and are used to minimize a “quality function” f(q). The quality function incorporates criteria based on energy, reference position, acceleration, and a scalar parameter p.

[0050] In the method according to the invention, a commanded end-effector destination shift Δx_c: =(Δx_t, Δx_r) is specified with the use of the Jacobi Matrix of kinematics at a time T_i, said end-effector destination shift being commanded by the operator with the aid of a space mouse or control ball in the form of a 6-dimensional increment vector. Δx_tε ³or Δx_rε ³always denote the translational or rotational portion of the commanded end-effector destination shift, whereas Δx_t^maxor Δx_r^max, respectively, each define the maximum (scalar-value) translational or rotational Cartesian position offset of the end effector per sampling time ΔT.

[0051] The desired Cartesian translational or rotational end-effector destination shift Δx_t^dor Δx_r^dper sampling time ΔT is defined as follows: $\Delta x_t^d:=\{\begin{array}{cc}\Delta x_t& \mathrm{if}\left|\Delta x_t\right|\le \Delta x_t^{\max }\\ \Delta x_t\frac{\left|\Delta x_t\right|}{\Delta x_t^{\max }}& \mathrm{otherwise}\end{array}\Delta x_r^d:=\{\begin{array}{cc}\Delta x_r& \mathrm{if}\left|\Delta x_r\right|\le \Delta x_r^{\max }\\ \Delta x_r\frac{\left|\Delta x_r\right|}{\Delta x_r^{\max }}& \mathrm{otherwise}\end{array}$

[0052] The value |x| of a vector x is determined here and hereinafter by its Euclidian norm $\left|x\right|:=\sqrt{\left(\sum _i{x}_i^2\right)},$

[0053] The desired end-effector destination shift is explained by Δx_d: =(Δx_t^d, Δx_r^d). The energy criteria, reference-position criteria and acceleration criteria can be suitably weighted according to the prevalent problem-specific requirements by allocating three positive numerical values α_j, β_j, γ_jto each axis j, which are predetermined by the operator, the value α_jserving in the weighting of an energy criterion that evaluates the difference between two adjacent, calculated articulation positions of axis j; and the value β_jserving in the weighting of a criterion that evaluates the deflection of the articulation position q_jfrom a reference value q_ref,j, which is also predetermined by the operator, and the value γ_jserving in the weighting of a criterion which evaluates the acceleration behavior in the form of an articulation speed offset of the articulation axis q_j. The articulation speed offset of the articulation axis q_jis predetermined by the operator, just like the physical articulation stops q_min, q_maxof the manipulator, which relate to path limitations, and the articulation speed limitations {dot over (q)}_maxand the articulation acceleration limitations {umlaut over (q)}_max. Since the method according to the invention is intended to optimize weighted criteria in the real-time cycle while guaranteeing that all path limitations are maintained, an allowable optimum articulation position offset Δqⁱis calculated for a desired end-effector destination shift Δx_dⁱat the time T_i−1during the time interval ΔT, with the algorithm of inverse kinematics that progresses during an initialization phase and a subsequent optimization phase. The current desired value is provided to the articulation regulators from q_sollⁱ⁺¹:=q_sollⁱ+Δqⁱ, with the articulation positions being shifted accordingly by the articulation regulators with q_sollⁱduring the time period ΔT. Allowable articulation position offset is understood to mean that the articulation position value that has been updated with Δq satisfies the physical articulation stops q_min, q_maxcorresponding to q_min≦q_soll+Δq_max, and Δq satisfies the articulation speed limitations {dot over (q)}_maxcorresponding to |Δqⁱ|≦{dot over (q)}_maxΔT. As a result, the limitations of the articulation accelerations are maintained reliably during the approach of articulation positions to the physical articulation stops. The type of optimum quality of Δqⁱcan be determined by the operator through the quality-criterion weighting α, β, γ. As a result, the sampling time ΔT corresponding to ΔT≧max (Δt_q, Δt_r) is dimensioned such that the computation time Δt_qfor calculating an optimum allowable articulation position offset Δq and the adjustment time Δt_r, which the regulators require for adapting the axial positions to the desired values q_soll, lie within the time span ΔT.

[0054] An algorithmic description of the method of the invention follows.

[0055] J_i, i=1, . . . , ndof denote the columns of the Jacobi Matrix of kinematics in the point q_iof the current articulation position of the manipulator, ndof denotes the number of articulations and ε_i:={dot over (q)}_maxⁱΔT denotes the maximum allowed articulation position offset per sampling time ΔT. With $y_i:=\frac{\Delta q^i}{{ϵ}_i}$

[0056] and J_ε:=(J₁ε₁, J₂ε₂, . . . , J_ndofε_ndof), the incremental kinematic equation is as follows:

[0057] pΔx_d=J_εy.  (1)

[0058] Because of the path limitations, the following box limitations result for y: $y_{\min }^i:=\max \left(-\frac{{\mathrm{bnd}}_{\mathrm{Low}}^i}{{ϵ}_i},\frac{q_{\min }^i-q_{\mathrm{soll}}^i}{{ϵ}_i}\right)$ $y_{\max }^i:=\min \left(-\frac{{\mathrm{bnd}}_{\mathrm{Up}}^i}{{ϵ}_i},\frac{q_{\max }^i-q_{\mathrm{soll}}^i}{{ϵ}_i}\right),{\mathrm{bnd}}_{\mathrm{Low}}^i:=\min \left({ϵ}_i,\frac{\left|q_{\min }^i-q_{\mathrm{soll}}^i\right|{\ddot{q}}_{\max }^i\Delta T^2}{{ϵ}_i}\right)$ ${\mathrm{bnd}}_{\mathrm{Up}}^i:=\min \left({ϵ}_i,\frac{\left|q_{\max }^i-q_{\mathrm{soll}}^i\right|{\ddot{q}}_{\max }^i\Delta T^2}{{ϵ}_i}\right).$

[0059] Further define: $y_{\mathrm{ref}}^i:=\frac{q_{\mathrm{ref}}^i-q_{\mathrm{soll}}^i}{{ϵ}_i},y_{\mathrm{acc}}^i:=\frac{\Delta {\stackrel{\_}{q}}^i}{{ϵ}_i},$

[0060] with Δ{overscore (q)}ⁱdenoting the recently calculated increment of the i-th articulation axis.

[0061] So that all path limitations can be met, according to the kinematic equation (1), the desired end-effector destination shift is subjected to a centric extension described in the form of a scalar:

[0062] 0≦p≦1.

[0063] The value p is maximized with the method of the invention such that all path limitations are maintained, with p·100 being the attained percentage of the desired end-effector destination shift.

[0064] Define the limitation vector b:=0, 0, 0, 0, 0, 0, y_min, 0,−y_max, −1) ε ²ⁿ⁺⁶, the parameter vector x: =(y,p) to be determined and the matrix of the gradients of all secondary conditions A ε ^2n+6,nwith n: =ndof+1: $\begin{array}{cc}A:=\left(\begin{array}{cc}{J}_{ϵ},& -\Delta x_d\\ I_n& \\ -I_n& \end{array}\right).& \left(2\right)\end{array}$

[0065] Here I_nε ^n,nindicates the unit matrix. Furthermore, α_i, i=1, . . . , 2n+6 indicate the lines of A.

[0066] The weights α, β, γ of the criteria define the Cholesky factor of the Hesse matrix from the scaled sum of the criteria in the form of the diagonal matrix Λε ^n,n. These criteria are the energy criterion (q−q_i)^tdiag(α_j) (q−q_i), the reference-position criterion (q−q_ref)^tdiag (β_j) (q−q_ref), with the value q_refbeing a predetermined articulation position value which is determined such that the sequence of the calculated articulation position values (q_i) runs near this reference position value; the acceleration criterion (q−2q_i+q_i−1)^tdiag(γ_j) (q−2q_i+q_i−1) and the additional criterion from −p, whereby the scalar parameter p satisfies the kinematic equation pΔx_d=J(q_i)(q−q_i) and the inequality 0≦p≦1; p·100 being the attained percentage of the commanded end-effector destination shift (Δx_d). The main diagonal elements of the above-mentioned diagonal matrix are: ${\Lambda }_{i,i}:=\sqrt{2\left({\alpha }_i^{\mathrm{scal}}+{\beta }_i^{\mathrm{scal}}+{\gamma }_i^{\mathrm{scal}}\right)}\mathrm{for}1\le i\le n-1$ $\mathrm{and}$ ${\Lambda }_{n,n}:=0$ $\mathrm{with}$ ${\alpha }_i^{\mathrm{scal}}:={\alpha }_i\frac{1}{\mathrm{ndof}},{\beta }_i^{\mathrm{scal}}:={\beta }_i\frac{{ϵ}_i}{\mathrm{ndof}\left(q_{\max }^i-q_{\min }^i\right)},{\gamma }_i^{\mathrm{scal}}:={\gamma }_i\frac{1}{3\mathrm{ndof}}.$

[0067] The base structure of the special method according to the invention, which utilizes an inverse kinematics for command control, is illustrated schematically in FIG. 2

[0068] An algorithm used in the method of the invention for the inverse kinematics is described in more detail below, in conjunction with FIG. 2. In the used algorithm of inverse kinematics, first an initialization phase is performed, which is configured as follows: Counting index: k=0; Starting value x_k=(y_k, p_k)=0; k max>2; Initialize with m_k: =7 the matrix of the gradients of the active secondary conditions {circumflex over (A)}^tε ^mn: ${\hat{A}}^t:=\left(\begin{array}{cc}{J}_{ϵ},& -\Delta x_d\\ 0,& 1\end{array}\right);$

[0069] initialize the index quantity J_k: =(j_k¹, j_k², . . . , j_k²ⁿ⁺⁶) to characterize the active, singular and inactive secondary conditions: $j_k^i:=\{\begin{array}{cc}n+6& \mathrm{for}i=7\\ i-1& \mathrm{for}i=8,\dots ,n+6\\ i& \mathrm{otherwise};\end{array}$

[0070] initialize the orthogonal triangular factorization of {circumflex over (A)}:

[0071] (r_k, J_k, Y_k, Z_k, L_k, S_k):=Φ₀^A({circumflex over (A)},m_k, J_k);

[0072] initialize the orthogonal regular triangular factorization of {overscore (Z)}:=ΛZ_k:

[0073] (Q_k^z,R_k^z):=Φ₀^z({overscore (Z)}r_k):

[0074] initialize the gradient vector of the quality function f(q) $\begin{array}{c}{g}_0\in {\mathrm{𝒯\Re }}^n:g_k:=2({\beta }_1^{\mathrm{scal}}\left(y_k^1-y_{\mathrm{ref}}^1\right)+{\gamma }_1^{\mathrm{scal}}\left(y_k^1-y_{\mathrm{acc}}^1\right)+\\ {\alpha }_1^{\mathrm{scal}}{y}_k^1,\dots ,{\beta }_{\mathrm{ndof}}^{\mathrm{scal}}\left(y_k^{\mathrm{ndof}}-y_{\mathrm{ref}}^{\mathrm{ndof}}\right)+\\ {\gamma }_{\mathrm{ndof}}^{\mathrm{scal}}\left(y_k^{\mathrm{ndof}}-y_{\mathrm{acc}}^{\mathrm{ndof}}\right)+{\alpha }_{\mathrm{ndof}}^{\mathrm{scal}}{y}_k^{\mathrm{ndof}},-1),\end{array}$

[0075] The following abbreviated form is used for g_k:

[0076] 2(β_i^scal(y_kⁱ−y_refⁱ)+γ_i^scal(y_kⁱ−y_accⁱ)+α_i^scaly_kⁱ−1),

[0077] In the used algorithm of inverse kinematics, an optimization phase is next, which is constructed as follows:

[0078] Calculate the direction of optimization d_k:

[0079] R_k^zd_aux=−Q_k^zZ_kg_k.

[0080] From this, determine d_auxthrough reverse substitution.

[0081] R_k^zd_z=Q_k^zd_aux.

[0082] From this, determine d_zthrough reverse substitution. Define the direction of optimization:

[0083] d_k:=Z_kd_z.

[0084] Determine the maximum step width s_kand the index j_kⁱ⁰of the limiting secondary condition: $s_k:=\{\begin{array}{cc}{\min }_j& \left\{\begin{array}{c}\frac{b_j-a_j^t{x}_k}{a_j^t{d}_k}\mathrm{with}{a}_j^t{d}_k<0\\ \mathrm{and}\forall i>m_k\mathrm{with}j=j_k^i\end{array}\right\}\text{=:}\frac{b_{\mathrm{j0}}-a_{\mathrm{j0}}^t{x}_k}{a_{\mathrm{j0}}^t{d}_k}\mathrm{with}{j}_k^0:=\mathrm{j0};\\ 1& \mathrm{if}{a}_j^t{d}_k\ge 0,\forall i>m_k\mathrm{with}j=j_k^i\end{array}$

[0085] (In the event of ambiguity, select the j0 having the smallest index i0).

[0086] III. Check for optimum quality and update all matrix and index values.

[0087] (a) If s_k<1 (secondary condition j_k⁰has become active):

[0088] i. If m_k<n (there are still inactive secondary conditions):

[0089] x_k+1=x_k+s_kd_k, x_k+1=(Y_k+1, P_k+1); g_k+1=2(β_i^scal(y_k+1ⁱ−y_refⁱ)+γ_i^scal(y_k+1ⁱ−y_accⁱ)+α_i^scaly_k+1ⁱ−1).

[0090] Update matrix factorization and index quantity: (r_k+1, m_k+1, J_k+1, Y_k+1, Z_k+1, L_k+1, S_k+1):=Φ₊^A(i0, r_k, m_k, J_k, Y_k, Z_k, L_k, S_k).

[0091] Check x_k+1, for optimum quality:

[0092] If Z_k+1^tg_k+1=0 and λ_k+1=LLS(r_k+1, m_k+1, g_k+1, Y_k+1, L_k+1)≧0 then: go to step IV.

[0093] Otherwise: k=k+1 (increase iteration index)

[0094] If k>k max: go to step IV.

[0095] Otherwise: Determine the orthogonal, regular triangular factorization of

[0096] {overscore (Z)}:=ΛZ_k+1:

[0097] (Q_k+1^z,R_k+1^z):=Φ^z({overscore (Z)},r_k+1)

[0098] Go to Step I.

[0099] ii. If m_k=n (corner of allowable range is attained, exchange of active secondary conditions is necessary): $x_{k+1}=x_k+s_k{d}_k,x_{k+1}=\left(Y_{k+1},P_{k+1}\right);$ $g_{k+1}=2\left({\beta }_i^{\mathrm{scal}}\left(y_{k+1}^i-y_{\mathrm{ref}}^i\right)+{\gamma }_i^{\mathrm{scal}}\left(y_{k+1}^i-y_{\mathrm{acc}}^i\right)+{\alpha }_i^{\mathrm{scal}}{y}_{k+1}^i,-1\right)$ ${\lambda }_k=\mathrm{LLS}\left(r_{k+1}{m}_{k+1},g_k,Y_k,L_k\right).$

[0100] Release a direction having the highest optimization potential: ${\lambda }_k^{\mathrm{j0}}:=\underset{j}{\min }{\lambda }_k^j;$

[0101] Update matrix factorizations and index quantity:

[0102] {tilde over (r)}_k,{circumflex over (m)}_k,{tilde over (J)}_k,{tilde over (Y)}_k,{tilde over (Z)}_k,{tilde over (L)}_k, {tilde over (S)}_k)=Φ₋^A(j0,r_k,m_k,J_k,Y_k,Z_k,L_k,L_k, S_k)

[0103] Activate secondary condition j_kⁱ⁰.

[0104] Update matrix factorizations and index quantity:

[0105] (r_k−1,m_mk+1,J_k+1,Y_k+1,Z_k+1,L_k+1,S_k+1):=Φ₊^A(i0,{tilde over (r)}_k,{tilde over (m)}_k,{tilde over (J)}_k,{tilde over (Y)}_k,{tilde over (Z)}_k,{tilde over (L)}_k,{tilde over (S)}_k).

[0106] Examine x_k+1for optimum quality:

[0107] If Z_k+1^tg_k+1=0; and λ_k+1=LLS(r_k+1,m_k+1,g_k+1,Y_l+1,L_k+1)≧0

[0108] then: go to step IV.

[0109] Otherwise: k=k+1 (increase iteration index)

[0110] If k>k max: go to step IV.

[0111] Otherwise: Determine the orthogonal, regular triangular factorization of

[0112] {overscore (Z)}:=ΛZ_k−1: (Q_k+1^z,R_k+1^z):=Φ^z({overscore (Z)},r_k+1).

[0113] Go to Step I.

[0114] If s_k=1 (no new active secondary condition).

[0115] x_k+1=x_k+d_k, x_k+1=(Y_k+1, P_k+1);

[0116] g_k+1=2(β_i^scal(y_k+1ⁱ−y_refⁱ)+γ_i^scal(y_k+1ⁱ−y_accⁱ)+α_i^scaly_k+1ⁱ−1);

[0117] Update matrix factorizations and index quantity:

[0118] (r_k+1, m_k+1, J_k+1, Y_k+1, Z_k+1, L_k+1, S_k+1):=(r_k, m_k, J_k, Y_k, Z_k, L_k, S_k).

[0119] Check x_k+1for optimum quality:

[0120] If Z_k+1^tg_k+1=0;

[0121] and λ_k+1=LLS(r_k+1, m_k+1, g_k+1, Y_k+1, L_k+1)≧0

[0122] then: go to step IV.

[0123] Otherwise: Release a direction having the highest optimization potential:

[0124] λ_k^j0:=_j^minλ_k^j;

[0125] Update matrix factorizations and index quantity:

[0126] (r_k+1, m_k+1, J_k+1, Y_k+1, Z_k+1, L_k+1, S_k+1):=Φ₋^A(j0, r_k, m_k, J_k, Y_k, Z_k, L_k, S_k)

[0127] −k=k+1 (increased iteration index)

[0128] If k>k max: go to Step IV.

[0129] Otherwise: Determine the orthogonal, regular triangular factorization of

[0130] {overscore (Z)}:=ΛZ_k+1:

[0131] (Q_k+1^z, R_k+1^z):=Φ^z({overscore (Z)},r_k+1).

[0132] Go to Step I.

[0133] IV. Solution x_k+1is determined. Stop!

[0134] End of algorithm of inverse kinematics.

[0135] Definition of the finction LLS:

[0136] λ=LLS(r, m, g, Y, L).

[0137] λ=0, i=r+1, . . . , m;

[0138] λ_nL:=(λ_r+1, . . . , λ_m);

[0139] L^tλ_L=y^tg.

[0140] From this, determine λ_Lε ^rthrough reverse substitution. Define the output value of the function:

[0141] λ:=(λ_L, λ_hL).

[0142] With l_eq:=1+max_l≦_i≦_m|λ(i)|, set the components of λ that are among the secondary conditions of Equation (1) at the value l_eq. End of function LLS.

[0143] Definition of the function Φ₊:

[0144] ({overscore (r)}, {overscore (m)},{overscore (J)},{overscore (Y)},{overscore (Z)},{overscore (L)},{overscore (S)})=Φ₊^A(i0,r,m,J,Y,Z,L,S).

[0145] Activate secondary condition jⁱ⁰, The condition is applied that α_j0with j0: =jⁱ⁰designate the jⁱ⁰-th line of A. Define Q^t:=(Y, Z) and form {overscore (α)}:=Qα_j0. Partition {overscore (α)}=:({overscore (α)}_y,{overscore (α)}_z) with {overscore (α)}_yε ^rand {overscore (α)}_zε ^n−r. Determine the householder reflection {tilde over (H)} ε ^n−r,n−rso that |{overscore (α)}_z|e₁={tilde over (H)}{overscore (α)}_z, with e₁:=(1, 0, . . . , 0)ε ^n−r. Define the unitary transformer: H ε ^n,naccording to: $H:=\left(\begin{array}{cc}{I}_r,& 0\\ 0,& \tilde{H}\end{array}\right)$

[0146] Define the unitary transformer:

[0147] {overscore (Q)}:=HQ;

[0148] Update indices: $\stackrel{\_}{r}:=\{\begin{array}{cc}r& \mathrm{if}\left|{\stackrel{\_}{a}}_z\right|=0\\ r+1& \mathrm{otherwise};\end{array}$

[0149] {overscore (m)}:=m+1;

[0150] Exchange secondary conditions:

[0151] If {overscore (r)}≠r, then ${\stackrel{\_}{j}}^i:=\{\begin{array}{cc}{j}^{\mathrm{i0}}& \mathrm{if}i=\stackrel{\_}{r}\\ j^{\stackrel{\_}{m}}& \mathrm{if}i=\mathrm{i0}\\ j^{\stackrel{\_}{r}}& \mathrm{if}i=\stackrel{\_}{m},\\ j^i& \mathrm{otherwise}\end{array}$

[0152] otherwise ${\stackrel{\_}{j}}^i:=\{\begin{array}{cc}{j}^{\mathrm{i0}}& \mathrm{if}i=\stackrel{\_}{m}\\ j^{\stackrel{\_}{m}}& \mathrm{if}i=\mathrm{i0}\\ j^i& \mathrm{otherwise};\end{array}$

[0153] {overscore (J)}:=({overscore (j)}¹, . . . , {overscore (j)}²ⁿ⁺⁶).

[0154] Update matrix factorizations.

[0155] {overscore (Y)} is the partial matrix of {overscore (Q)}^tthat includes columns 1 through {overscore (r)} of Q{overscore (Q)}^t.

[0156] {overscore (Z)} is the partial matrix of {overscore (Q)}^tthat includes columns {overscore (r)}+1 through n of {overscore (Q)}^t. $\stackrel{\_}{L}:=\left(\begin{array}{cc}L,& 0\\ {\stackrel{\_}{a}}_y& \left|{\stackrel{\_}{a}}_z\right|\end{array}\right);$ $\stackrel{\_}{S}:=\{\begin{array}{cc}\left(S,{\stackrel{\_}{a}}_y\right)& \mathrm{if}\stackrel{\_}{r}=r\\ S& \mathrm{otherwise}.\end{array}$

[0157] End of the function Φ₊.

[0158] Definition of the function Φ.:

[0159] ({overscore (r)},{overscore (m)},{overscore (J)},{overscore (Y)},{overscore (Z)},{overscore (L)},{overscore (S)})=Φ₋^A(i0,r,m,J,Y,Z,L,S).

[0160] Inactivate secondary condition jⁱ⁰.

[0161] If j0>r: (Eliminate singular, active secondary condition) Update indices:

[0162] {overscore (r)}:=r;

[0163] {overscore (m)}:=m−1;

[0164] Exchange secondary conditions: ${\stackrel{\_}{j}}^i:=\{\begin{array}{cc}{j}^{\mathrm{i0}}& \mathrm{if}i=\stackrel{\_}{m}+1\\ j^{\stackrel{\_}{m}+1}& \mathrm{if}i=\mathrm{j0}\\ j^i& \mathrm{otherwise}\end{array}.$

[0165] {overscore (J)}:=({overscore (j)}¹, . . . , {overscore (j)}²ⁿ⁺⁶).

[0166] Update matrix factorizations:

[0167] {overscore (Y)}:=Y;

[0168] {overscore (Z)}:=Z;

[0169] {overscore (L)}:=L;

[0170] {overscore (SL)}:=(S₁, . . . ,S_j0−r−l,S_j0−r+1,S_m−r),

[0171] where S_idenote the columns of S.

[0172] If j0≦r: (Eliminate regular active secondary condition) {tilde over (R)} results from R:=L^tthrough striking of the j0-th column. The elements of {tilde over (R)} that occur at the locations (j0,j0+1), (j0+1, j0+2), . . . , (r,r+1) are canceled through left manipulation with a sequence of unitary elimination matrices {tilde over (E)}_j0,j0+1, . . . , {tilde over (E)}_r−1,rεIR^r,r. Define linear transformers:

[0173] {circumflex over (R)}:={tilde over (E)}_r−1,r{tilde over (E)}_r−2,r−1. . . {tilde over (E)}_j0,j0+1{tilde over (R)};

[0174] {circumflex over (S)}:={tilde over (E)}_r−1,r{tilde over (E)}_r−2,r−1. . . {tilde over (E)}_j0,j0+1S; $E_{i,i+1}:=\left(\begin{array}{cc}{\tilde{E}}_{i,i+1},& 0\\ 0& I_{n-r}\end{array}\right);$

[0175] Q^t:=(Y,Z);

[0176] {overscore (Q)}:=E_r−1,rE_r−2,r−1. . . E_j0,j0+1Q;

[0177] If the last line of {circumflex over (S)} is zero, or if r=m:

[0178] Update indices:

[0179] {tilde over (r)}:=r−1;

[0180] {overscore (m)}:=m−1;

[0181] Exchange secondary conditions: ${\stackrel{\_}{j}}^i:=\{\begin{array}{cc}{j}^{\mathrm{i0}}& \mathrm{if}i=\stackrel{\_}{m}+1\\ j^{i+1}& \mathrm{if}i=\mathrm{j0},\dots ,\stackrel{\_}{m}\\ j^i& \mathrm{otherwise}\end{array}.$

[0182] {overscore (J)}:=({overscore (j)}¹, . . . , {overscore (j)}²ⁿ⁺⁶).

[0183] Update matrix factorizations:

[0184] {overscore (Y)} is the partial matrix of {overscore (Q)}^tthat includes columns 1 through {overscore (r)} of {overscore (Q)}^t.

[0185] {overscore (Z)} is the partial matrix of {overscore (Q)}^tthat includes columns {overscore (r)}+1 through n of {overscore (Q)}^t.

[0186] If r<m, strike the last line of {circumflex over (S)} and {circumflex over (R)}. Set:

[0187] {overscore (L)}:={circumflex over (R)}^t

[0188] {overscore (S)}:={circumflex over (S)};

[0189] Otherwise (convert singular, active secondary condition into a regular, active secondary condition):

[0190] Determine the element of the last line of {circumflex over (S)} with the smallest index l0 such that:

[0191] {circumflex over (S)}_r,i0≠0.

[0192] Exchange column l0 with column 1 of {circumflex over (S)}. Define linear transformer:

[0193] {overscore (R)}:=({circumflex over (R)}, {circumflex over (S)}_l0),

[0194] where {circumflex over (S)}_l0denotes the l0th column of {circumflex over (S)}.

[0195] Update indices:

[0196] {overscore (r)}:=r;

[0197] {overscore (m)}:=m−1;

[0198] Exchange secondary conditions: ${\hat{j}}^i:=\{\begin{array}{cc}{j}^{\mathrm{j0}}& \mathrm{if}i=\stackrel{\_}{m}+1\\ j^{i+1}& \mathrm{if}i=\mathrm{j0},\dots ,\stackrel{\_}{m}\\ j^i& \mathrm{otherwise}\end{array}.{\stackrel{\_}{j}}^i:=\{\begin{array}{cc}{\hat{j}}^{\stackrel{\_}{r}+\mathrm{l0}-1}& \mathrm{if}i=\stackrel{\_}{r}\\ {\hat{j}}^{\stackrel{\_}{r}}& \mathrm{if}i=\stackrel{\_}{r}+\mathrm{l0}-1\\ {\hat{j}}^i& \mathrm{otherwise}\end{array}$

[0199] {overscore (J)}:=({overscore (j)}¹, . . . , {overscore (j)}²ⁿ⁺⁶).

[0200] Update matrix factorizations:

[0201] {overscore (Y)} is the partial matrix of {overscore (Q)}^tthat includes columns 1 through {overscore (r)} of {overscore (Q)}^t.

[0202] {overscore (Z)} is the partial matrix of {overscore (Q)}^tthat includes columns {overscore (r)}+1 through n of {overscore (Q)}^t.

[0203] {overscore (L)}:L={overscore (R)}^t;

[0204] {overscore (S)}:=({circumflex over (S)}₂, . . . , {circumflex over (S)}_l0−1, {circumflex over (S)}₁, {circumflex over (S)}_l0+1, . . . , {circumflex over (S)}_m−r),

[0205] where {circumflex over (S)}₁denotes the columns of {circumflex over (S)}.

[0206] End of function Φ.

[0207] Definition of the function Φ₀^A:

[0208] ({overscore (r)},{overscore (J)},{overscore (Y)},{overscore (Z)},{overscore (L)},{overscore (S)})=Φ₀^A({circumflex over (A)},m,J).

[0209] 1. Define:

[0210] A⁰:={circumflex over (A)}, i:=0.

[0211] 2. Define the following matrix recursion:

[0212] Aⁱ⁺¹L=H_iAⁱ, i≧0.

[0213] The unitary transformer H_iε ^n,nis explained as follows: $H_i:=\left(\begin{array}{cc}{I}_i& 0\\ 0,& {\tilde{H}}_i\end{array}\right).$

[0214] The householder reflection {tilde over (H)}_iε ^n−i,n−1is defined such that:

[0215] |{tilde over (α)}_k0ⁱ|e₁={tilde over (H)}_i{tilde over (α)}_k0ⁱ,

[0216] with e₁:=(1, 0, . . . , 0) ε ⁿ⁻ⁱ. Here the vectors {tilde over (α)}_kⁱεⁿ⁻ⁱ, k=1, . . . , n−i indicate the columns of the matrix {tilde over (A)}ⁱε ^n−i,m−1: $A^i:=\left(\begin{array}{cc}*,& *\\ *,& {\tilde{A}}^i\end{array}\right)$

[0217] Moreover, 1≦k0≦n−i is the smallest index with {tilde over (α)}_k0ⁱ≠0. If no such k0 exists, then go to step III; otherwise go to step IV.

[0218] 3. Define indices:

[0219] {overscore (r)}:=i;

[0220] {overscore (J)}:=({overscore (j)}ⁱ, . . . , {overscore (j)}²ⁿ⁺⁶).

[0221] Define matrix factorizations:

[0222] {overscore (Q)}:=H_{{overscore (r)}−1}H_{{overscore (r)}−2}. . . H₀;

[0223] {overscore (Y)} is the partial matrix of {overscore (Q)}^tthat includes columns 1 through {overscore (r)} of {overscore (Q)}^t.

[0224] {overscore (Z)} is the partial matrix of {overscore (Q)}^tthat includes columns {overscore (r)}+1 through n of {overscore (Q)}^t.

[0225] T is the matrix that results from the striking of lines +1 through n from Aⁱ⁺¹.

[0226] Define {overscore (L)} ε ^{{overscore (r)},{overscore (r)}} and {overscore (S)}ε ^{{overscore (r)},m−{overscore (r)}} according to:

[0227] ({overscore (L)}^t,{overscore (S)}):=T.

[0228] Stop: function Φ₀^Ahas been performed completely!

[0229] 4. Exchange column i+k0 with column i+1 in Aⁱ. ${\stackrel{\_}{j}}^k:=\{\begin{array}{cc}{j}^{i+\mathrm{k0}}& \mathrm{if}k=i+1\\ j^{i+1}& \mathrm{if}k=i+\mathrm{k0}\\ j^k& \mathrm{otherwise}\end{array}$

[0230] Set j^k: ={overscore (j)}^k∀k, increase the counting index i=i+1 and go to Step II.

[0231] End of function Φ₀^A.

[0232] Definition of the function Φ₀^Z:

[0233] (Q^z, R^z):=Φ₀^z({overscore (Z)},r).

[0234] Determiine the QR triangular factorization of {overscore (Z)}: $Q^z\stackrel{\_}{Z}=\left(\begin{array}{c}{R}^z\\ 0\end{array}\right).$

[0235] Here Q^zε ^n,ndesignate a unitary matrix, and R^zε ^r,ran upper triangular matrix.

[0236] End of function Φ₀^z.