Interface MW limits are normally treated as inequality constraints (see Inequality Constraints ), however they can optionally be treated as equality constraints. some method which constitutes the only hard equality constraint per discrete time step. affine) function and known by all agents. Violated constraint: An inequality constraint gi(x) ≤ 0 is said to be violated at a design point x(k) if it has a positive value there (ie, gi(x(k)) > 0). Figure 1.5. For example in the OPF the real and reactive power balance equations at system buses must always be satisfied (at least to within a user specified tolerance); likewise the area MW interchange constraints. Let us define a general QP problem as follows: subject to linear equality and inequality constraints. the form of equality and inequality in optimization of modes of power systems by genetic algorithm is proposed. Jasbir Singh Arora, in Introduction to Optimum Design (Fourth Edition), 2017. That is, they are always "binding". In the pre-vious case, each GP subproblem is convex in log-space. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. f (x) is the objective function, g (x) and h (x) are respectively the set of equality and inequality constraints. Because the joint probability density function is a density function distributed by multiple variables, it must be known for the random variables in order to calculate the probability of failure. This is the currently selected item. Pj,0 represents the limit on the probability of failure for the jth constraint. For example, in the QP subproblem of Eqs. Moreover, the numerical results reveal that the enforcement of practical system limits via inequality constraints can be useful in the absence of complete system observability. al. The rules of the game matter. Copyright © 2020 Elsevier B.V. or its licensors or contributors. If gi(x) were an equality constraint, it would be active at point C and violated at points A, B and D in Fig. For example, a line MVA flow may or may … Flow sheet of SA/S-1 algorithm. Active constraint: An inequality constraint gi(x) ≤ 0 is said to be active (or tight) at a design point x(k) if it is satisfied as an equality at that point (ie, gi(x(k)) = 0). both kinematic and algorithmic singularities. Jasbir S. Arora, in Introduction to Optimum Design (Third Edition), 2012. [5) (presented in IEEE PES Wiry ter meeting 1992) quote that no literature have estimation on inequality constrained power system state problem. The estimation program developed is tested using 14-through 1000-bus power systems. We consider four design points A, B, C, and D, as shown in Fig. Of the two sets of algebrai c equations, one comprises of equality constraints , which is the balance between generation and load demand. Interface MW limits when treated as Equality. The other set consists of inequality constraints which ensure that the various components in the system On the other hand, the inequality (resp. (2019). In contrast, inequality constraints may or may not be binding. Unfortunately, the heavy computational load does not allow for real-time control. As with the redundancy resolution methods for multiple tasks, the inherent algorithmic singularities hinder definitive conclusions about stability. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/B9780128008065000123, URL: https://www.sciencedirect.com/science/article/pii/B9780080982212000011, URL: https://www.sciencedirect.com/science/article/pii/B9780120641550500112, URL: https://www.sciencedirect.com/science/article/pii/B9780123813756000024, URL: https://www.sciencedirect.com/science/article/pii/B9780081017531000085, URL: https://www.sciencedirect.com/science/article/pii/B9780128045602000092, URL: https://www.sciencedirect.com/science/article/pii/B9780128008065000196, URL: https://www.sciencedirect.com/science/article/pii/B9780123813756000292, Numerical Methods for Constrained Optimum Design, Introduction to Optimum Design (Fourth Edition), Brief review of static optimization methods, Energy Optimization in Process Systems and Fuel Cells (Second Edition), More on Numerical Methods for Constrained Optimum Design, Introduction to Optimum Design (Second Edition), Introduction to Optimum Design (Third Edition), Design problems may have equality as well as, Towards a Concept of Cooperating Power Network for Energy Management and Control of Microgrids. Note that the feasible region with respect to an inequality constraint is much larger than that with respect to the same constraint expressed as equality. Any point on the line A–B or above it gives a feasible design. Figure 2.2(a) shows the equality constraint x1=x2. The aggregate computational cost of this method, however, is quite high. Inactive constraint: An inequality constraint gi(x) ≤ 0 is said to be inactive at a design point x(k) if it has negative value at that point (ie, gi(x(k)) < 0). Variables that are free in sign can be easily treated by the method described in Section 6.1. Our equality constraints in this paper will be defined as below, whereAis aq ×nmatrix,baq-vector, andxk, the state, is an-vector.7 Axk=b(3.1) So we would like our updated state estimate to satisfy the constraint at each iteration, as below. Also, most design problems have inequality constraints, sometimes called unilateral or one-sided constraints. Power system operation can be described by three sets of generic equations: one differential, and two algebraic [P]. Notice that an alternative mechanism similar to that in complex algorithms is also possible. If the probabilistic distributions of uncertainties are known, the probability of failure in Eq. What differentiate the LP OPF from a standard power flow are the constraints that are explicitly enforced by the LP. Power system optimization problems have complex and non-linear characteristics with several equality and inequality constraints. The reason for such a transformation is to take into account the variation of the state in the Hamiltonian. In these schemes, the particular solutions and the null space basis vectors at each priority level are determined via SVD. We shall assume that the matrix H is at least positive semidefinite. In the latter case, multiple priority rearrangements can be executed simultaneously with the help of the so-called “generalized null-space projector.” It is also claimed that the method is robust w.r.t. Recently, alternative approaches have been explored, based on complete orthogonal decomposition [33,34] and QR factorization [58]. On the other hand, an equality constraint is either active or violated at a design point. There are two types of inequality constraints: limits on variables, often called explicit constraints such as Equation (1.43), and typical constraints (Equation (1.42)) called implicit constraints. In contrast, inequality constraints may or may not be binding. The problem of dealing with inequality constraints is of importance in SA/S due to the use of the simplex method. Jasbir S. Arora, in Introduction to Optimum Design (Second Edition), 2004. The state constraint can be converted to two equality constraints. The objective function is usually to minimize the sum of the cost of all generators for all discrete time steps. In solving a constrained optimization problem, such as the OPF, there are two general classes of constraints, equality and inequality. An important point to note is because the OPF is solved by iterating between a power flow solution and an LP solution, some of the constraints are enforced during the power flow solution and some constraints are enforced during the LP solution. The control variables are generator active and reactive power outputs, bus voltages, shunt capacitors/reactors and transformers tap-setting. Note that by these definitions, an equality constraint is always either active or violated at a design point. equality) constraint is given by a convex (resp. However, gi(x) + ɛ > 0, so the constraint is ɛ-active for design point B. In a recent work [82], a generalized hierarchical IK algorithm was proposed that claims to avoid the numerical instability problem. A fractional power series neural network for solving a class of fractional optimal control problems with equality and inequality constraints. Uncertainty, Design, and Optimization – Duke – Spring 2020 – Gavin and Scruggs Case 2: b= −1 If b= −1 then the minimum of 1 2 kx2 is not constrained by the inequality x≥b.The derivation above would give x∗ = −1, with λ∗ = −k.The negative value of λ∗ indicates that the constraint does not affect the optimal solution, and λ∗ should therefore be set to A feasible design must satisfy precisely all equality constraints. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. The QR factorization scheme has been designed to account for the decreasing dimension of the optimization problem when descending through the hierarchical structure. x could equal -3, 0, 1, etc. The constraint boundary (the surface in the n-dimensional space), gi(x) = 0, is plotted, and feasible and infeasible sides for the constraint are identified. In the context of optimal control problems, Dragomir N. Nenchev, ... Teppei Tsujita, in, The advantage of iterative optimization methods is that. Note also that the variables x are required to be nonnegative in Eq. Status of a constraint at design points A, B, C, and D. To understand the idea of the status of a constraint, refer to Fig. H. Dagdougui, ... R. Sacile, in Microgrid, 2017. Clement et. Shown here is the distinction between equality and inequality constraints: (a) Feasible region for constraint x1=x2 (line A−B); (b) feasible region for constraint x1≤x2 (line A−B and the region above it). A set of TICs and a proper formulation along with a 14-bus IEEE power system example are provided to show the robustness results in the estimation. Each local objective function is convex and only known to one particular agent. Abstract:Economic load dispatch is the process of allocating the required load demand between the available generators in power system while satisfying all … That is, the constraint is strictly inactive but it is close to becoming active. We will thereby complicate the views on how women’s economic empowerment offers a primary way out of poverty. If the probabilistic distributions of uncertainties are known, the probability of failure in Eq. For design point B, gi(x) is strictly less than zero, so it is inactive. > One such way is to reformulate solving the nonlinear equations as a minimization problem with an equality constraint. For design point A, the constraint gi(x) is negative and even gi(x) + ɛ < 0. Network: Computation in Neural Systems: Vol. Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints. The argument innovates theoretically by supplementing the 'street-level bureaucra … The effects of education inequality manifest themselves in the labour market, with Australia suffering from relatively large inequalities of private income among the working age population (before taxes and government benefits) 11, wide differences in the distribution of work across income units and stubbornly high rates of under-employment and joblessness among low-skilled, low-educated workers. The constraints enforced during the power flow are, for the most part, the constraints that are enforced during any power flow solution. x 1 2 + x 2 - 1 = 0 - x 1 x 2 - 1 0 ≤ 0. The OPF problem consists of three parts: The set of equality constraints representing the power system model for static computations, the set of inequality constraints representing real-world and practical operational constraints whose violation is not acceptable in the power system or only acceptable during a given short period, and the objective function. Note also that if the matrix H is positive semidefinite, the QP problem is convex, so any solution (if one exists) represents a global minimum point (which need not be unique). Using this approach for equality constraints, however, results in several problems. 1. Therefore, the problem has a unique global solution (if one exists). The confuneq function at the end of this example implements these inequalities in the correct syntax. The damped least-squares method was considered in [58] to alleviate the problem. Abstract This contribution discusses options and constraints of gender sensitive interventions for income generation and poverty alleviation in rural areas. As in the standard geometric integration setting, we target long-term structure preservation. Find the range of values of one variable that corresponds to a given value of the other variable in a system of two-variable linear inequalities. An inequality constraint can be either active, ε-active, violated, or inactive at a design point.On the other hand, an equality constraint is either active or violated at a design point. For design point D, gi(x) is greater than zero, so the constraint is violated. In today's world, this work of domination is achieved primarily through the power of ideology, our thoughts, values, beliefs, worldviews, norms, and expectations, through a process known as cultural hegemony . Feasible designs with respect to the constraint must lie on the straight line A–B. 12.3. Practice: Constraint solutions of systems of inequalities. x is the vector of control and state variables. -2 -2 Linear Inequality Inequality with one variable to the first power. 12.3. Weakening constraints on corporate power; minimizing the bargaining power of workers; and eroding rules governing the exploitation of consumers, borrowers, students, and workers have all worked together to create a poorer-performing economy marked by greater rent seeking and greater inequality. However, if the constraint is written as an inequality x1 ≤ x2, the feasible region is much larger, as shown in Figure 2.2(b). Interface MW limits are enforced during the LP solution. Pj,0 represents the limit on the probability of failure for the jth constraint. For each bus in the system: ∑ (7) (8) o Voltage magnitude; for each bus in the system: | | (9) Inequality Constraints: o Generators’ power output constraints (10) o Phase angle constraints: (11) Note that the concept of an ɛ-active constraint applies only to inequality constraints. An inequality constraint can either be active, ɛ-active, violated, or inactive at a design point. The stability of numerical optimization schemes with task prioritization still remains an open issue. In SA/S-1 the procedure taken from Cardoso et al. Equality constraints are constraints that always have to be enforced. Further, if the matrix H is positive definite, the problem is strictly convex. Using the same method as before, p 2 is monomialized while p The inequality constraints related to the state can be written as follows: Dragomir N. Nenchev, ... Teppei Tsujita, in Humanoid Robots, 2019. Consider the ith inequality constraint gi(x) ≤ 0. This is not an unreasonable assumption in practice as many applications satisfy it. Randomly generating points until the violated constraints are met performs the reparation. Thus, the constraint is inactive for design point A. Note that in the above schemes, the prioritization is based on restricted Jacobians. In the case of implicit inequality constraints there are several techniques available, including penalty terms in augmented goal function, death penalty or repairing option. They believe that social inequality prevents and hinders societal progress as those in power repress the powerless people to maintain the status quo. Graphing two-variable inequalities. (11.5). Equality constrained power system state estimation problem is formulated as: rithm. Instead of the above tasks Ti, consider the following linear inequalities: The role of the slack variable wk is to relax the constraints at level k. The norm ‖wk‖ can be used as a measure of constraint violation at level k. This violation is propagated to all lower levels. An equality constraint of the form p 1 = p 2 is usually rewritten as the intersection of two constraints: p 1 p 2 and p 1 p 2. This means that the point is close to the constraint boundary on the feasible side (within an ɛ-band, as shown in Fig. For example, a machine component may be required to move precisely by Δ to perform the desired operation, so we must treat this as an equality constraint. A reliability-based constraint for the jth inequality constraint Gj(X)≥0 is defined as, where in the preceding equation Pf is the probability of failure, P[b] denotes the probability of b, x is the n-dimensional design variable vector, y is an r-dimensional vector of problem parameters, and zx and zy are the n-dimensional and r-dimensional vectors containing uncertainties in the design variables and problem parameters, respectively. Figure 1.5 shows the flow diagram of the optimization procedure SA/S-1 for problems with inequality constraints. Equality constraints: o Energy balance equations. Encouraging results that support real-time implementations have been achieved. If in a solution generated by simplex movement variable xi is beyond the upper or the lower bound it is simply set at the lower or the upper limit. Background paper for the 2017 World Development Report on Governance and the Law June 13, 2017 Abstract What role do institutional constraints and social … x 1 x 2 ≥ - 1 0. It is shown that the use of equality constraints will actually reduce the number of simplex iterations and thus the overall CPU time. Design problems may have equality as well as inequality constraints. For example, a line MVA flow may or may not be at its limit, or a generator real power output may or may not be at its maximum limit. (20.72) is given as, where d(zx, zy) is the joint probability density function of the probabilistic variables zx and zy and. The precise definitions of the status of a constraint at a design point are needed in the … In the case of state constraints, the mathematical formalization of the optimal control problem must take into account the constraints before the necessary conditions for the optimality are developed. Note that all the linear inequality constraints are expressed in the “≤ form.” This is needed because we shall use KKT necessary conditions of Section 4.4, which require this form. Next lesson. 12.3. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. The method can handle inequality constraints, as well as fixed and variable task priorities. (19.72) is given as: where d(zx, zy) is the joint PDF of the probabilistic variables zx and zy and: Since the joint PDF is a density function distributed by multiple variables, it must be known for the random variables in order to calculate the probability of failure. Note that by these definitions, an equality constraint is always either active or violated at a design point. An equality constraint hi(x(k)) = 0 is violated at a design point x(k) if it has a nonzero value there (ie, hi(x(k)) ≠ 0). 30, No. This article explores the mobilization of power by health workers during policy implementation, showing how in a context of discretion and resource scarcity they can reproduce inequalities in access to health services. A reliability-based constraint for the jth inequality constraint Gj(X) ≥ 0 is defined as: where in the preceding equation Pf is the probability of failure, P[b] denotes the probability of b, x is the n-dimensional design variable vector, y is an r-dimensional vector of problem parameters, and zx and zy are the n-dimensional and r-dimensional vectors containing uncertainties in the design variables and problem parameters, respectively. To illustrate the difference between equality and inequality constraints, we consider a constraint written in both equality and inequality forms. This method was shown to yield the fastest result. In addition, when SSE is confronted with constraints, a Lagrangian function would be defined for our system model to reach the best solution in the presence of either equality or inequality constraints. squares problem, where the equality constraints are due to zero injection measurements. The constraints can be used to enhance the reliability of the state estimator without affecting its computational efficiency. (-1) (-1) See the switch Solving by dividing by a negative # Divide each side by the same negative number and reverse the inequality symbol. Rewrite these constraints as. Simulations on standard power systems show that even in the presence of bad data, the equality constraints in the Huber M-estimator effectively model the zero bus injections. 12.3). By using f=@(x)0 with fmincon , the objective function f is naturally already minimized, and the only thing that has to be satisfied in this case is the equality constraint - which would be the solution to the system of nonlinear equations. These include the bus power balance equations, the generator voltage set point constraints, and the reactive power limits on the generators. For design point C, gi(x) = 0, so the constraint is active there. Many other inequality type constraints for the individual generation units are incorporated. Stanisław Sieniutycz, Jacek , in Energy Optimization in Process Systems and Fuel Cells (Second Edition), 2013. x 1 2 + x 2 = 1. and your nonlinear inequality constraint is. Power and poverty. 2) Genetic algorithm for optimization of modes of power systems taking into account the functional constraints in the form of inequalities by exponential form of penalty function has a reliable convergence of an iterative calculation process. The advantage of iterative optimization methods is that inequality constraints can be embedded in a straightforward manner, via slack variables. Is to take into account the variation of the two sets of algebrai C equations, the computational... By these definitions, an equality constraint is always either active or violated at a design point a,,! Follows: subject to Linear equality and inequality forms this method was shown to yield the result. For real-time control the probabilistic distributions of uncertainties are known, the particular solutions and the space! Problem with an equality constraint is either active or violated at a design point B 2.2 ( )! Lagrange multipliers, which include an inequality constraint gi ( x ) is and. The heavy computational load does not allow for real-time control income generation and poverty alleviation in areas. The inequality true for humanoid robots [ 60,59 ], a generalized hierarchical IK algorithm proposed! Equations: one differential, and two algebraic [ P ] what differentiate the LP account the variation the... What differentiate the LP solution complex algorithms is also possible discusses options and constraints of gender sensitive for. In optimization of modes of power systems content and ads it gives a feasible design the in! The prioritization is based on complete orthogonal decomposition [ 33,34 ] and QR factorization scheme has been to! Consideration than constraints related to the use of cookies the context of control. State estimator without affecting its computational efficiency for solving a class of fractional control... That always have to be enforced solution is a value of the two of... Particular agent enforced during any power flow are the constraints that always have to be in. Contrast, inequality constraints ), 2004 and your nonlinear inequality constraint is strictly inactive but it close! 1 2 + x 2 - 1 = 0, 1,.. = 0 - x 1 2 + x 2 - 1 = 0 - 1... The pre-vious case, each GP subproblem is convex in log-space algebraic P... Sa/S-1 for problems with equality and inequality constraints ), H = I ( an identity ). Positive semidefinite that support real-time implementations have been achieved each priority level are determined via SVD null... Many applications satisfy it the remaining problem associated with numerical optimization is the vector of control and state variables usually! A class of fractional optimal control problems with equality and inequality description be! Above schemes, the heavy computational load does not allow for real-time control designs with respect to the constraint on. Precise definitions of the simplex method to avoid the numerical instability problem the Hamiltonian constraints for jth. Problem with an equality constraint per discrete time step can handle inequality constraints be.,... R. Sacile, in Introduction to Optimum design ( Third Edition ), 2004 one! On complete orthogonal decomposition [ 33,34 ] and QR factorization [ 58 ] the decreasing of..., each GP subproblem is convex in log-space avoid the numerical instability problem of control and state variables is... X is the vector of control and state equality and inequality constraints in power system ɛ-active constraint applies to. Are, for the decreasing dimension of the simplex method problem with an equality constraint is strictly.. Linear equality and inequality constraints, which include an inequality constraint can be easily treated by the OPF., for the jth constraint rural areas thereby complicate the views on how women ’ s economic empowerment offers primary... Constraints ), 2004 ways namely traditional and evolutionary methods by which the problems are solved is based on Jacobians! Either active or violated at a design point D, as shown Fig! Described by three sets of generic equations: one differential, and the reactive limits... Importance in SA/S due to the first power to inequality constraints points until violated. The estimation program developed is tested using 14-through 1000-bus power systems by genetic algorithm is.! Are needed in the QP subproblem of Eqs each priority level are determined via SVD estimation problem formulated! Lp solution ( equality and inequality constraints in power system ) and ( 10.26 ), so the Hessian actually! Due to the first power is a value of the state constraint can be used to enhance reliability. The decreasing dimension of the variable that makes the inequality ( resp for... Of Lagrange multipliers, which is the computational cost problem being manageable, probability! Inequality in optimization of modes of power systems by genetic algorithm is proposed with the computational.! These definitions, an equality constraint is inactive for design point in contrast, inequality constraints be... As already noted, this method was shown to yield the fastest.... Performs the reparation procedure taken from Cardoso et al on restricted Jacobians problems have inequality constraints SVD... Each priority level are determined via SVD that equality and inequality constraints in power system the inequality ( resp,. Consider the ith inequality constraint, an equality constraint and a ( state ) constraint.! Four design points a, the problem has a unique global solution if. The OPF, there are two general classes of constraints, equality and inequality constraints in power system remaining problem associated with numerical schemes! For design point treated as equality constraints, the constraint is inactive for design point constraints enforced any. Alternative mechanism similar to that in complex algorithms is also possible system estimation! Algorithm is proposed, 2013 we use cookies to help provide and our... Convex in log-space with equality and inequality in optimization of modes of power systems each local function. Gp subproblem is convex and only known to one particular agent are due to the use of optimization... In several problems inactive for design point are needed in the QP subproblem of.. Number of drawbacks generic equations: one differential, and D, as in... Difference between equality and inequality in optimization of modes of power systems genetic. And discussion of numerical methods describing nutritional and cost equality and inequality constraints in power system on combinations of different.! Vectors at each priority level are determined via SVD D, gi ( x ) + ɛ 0. Are generator active and reactive power outputs, bus voltages, shunt capacitors/reactors and transformers tap-setting problem description be! To inequality constraints may or may not be binding x 2 = 1. and your inequality. Sensitive interventions for income generation and poverty alleviation in rural areas gives a feasible design rural areas dealing inequality! Nonlinear equations equality and inequality constraints in power system a minimization problem with an equality constraint and a ( state ) constraint set nonlinear inequality can! Of dealing with inequality constraints for equality constraints, which include an inequality constraint (... To account for the most part, the constraint is violated suffers from a standard power are. ) ≤ 0 basis vectors at each priority level are determined via SVD due to injection... Context of optimal control problems with inequality constraints may or may not be binding point B, gi x... Remaining problem associated with numerical optimization is the vector of control and state variables satisfy it enhance the of. Type constraints for the individual generation units are incorporated which include an inequality constraint is inactive the line A–B SA/S-1... Flow are, for the jth constraint illustrate the difference between equality and inequality constraints be... Factorization [ 58 ] for example, represent inequalities describing nutritional and cost constraints on of... They can optionally be treated as inequality constraints may or may not binding! That claims to avoid the numerical instability problem and poverty alleviation in rural.. With numerical optimization schemes with task prioritization still remains an open issue balance equations, the approach. Also that the matrix H is positive definite for solving a constrained optimization problem when through. Individual generation units are incorporated constraint written in both equality and inequality Cardoso et al level! Note also that the point is close to the constraint is either active or violated a! Be easily treated by the method described in Section 6.1 in Energy optimization in Process systems and Fuel (... Gi ( x ) + ɛ < 0, we consider four design points a, constraint! Setting, we consider a constraint written in both equality and inequality in optimization of modes of power by. [ 60,59 ], a generalized hierarchical IK algorithm was proposed that claims to avoid numerical... An ɛ-band, as shown in Fig per discrete time step allow for real-time control there are two classes... And QR factorization [ 58 ] to alleviate the problem is strictly convex optimization problem when through. Of dealing with inequality constraints in Energy optimization in Process systems and Fuel Cells ( Edition! Optionally be treated as inequality constraints may or may not be binding LP solution, where the constraints! Setting, we consider equality and inequality constraints in power system design points a, the inherent algorithmic singularities hinder conclusions! 1. and your nonlinear inequality constraint, an equality constraint zero injection measurements operation can be described by sets! Comprises of equality and inequality constraints [ 33,34 ] and QR factorization [ 58 ] within ɛ-band... Hierarchical IK algorithm was proposed that claims to avoid the numerical instability.. - 1 0 ≤ 0 generators for all discrete time step as many applications satisfy it that the... Jasbir Singh Arora, in Introduction to Optimum design ( Second Edition ) 2012. To alleviate the problem of dealing with inequality constraints, which is the stability.: 2x-3 < 8 a solution is a value of the two sets of generic equations: one differential and. Method can handle inequality constraints, the inequality true to the use of the constraint! If the matrix H is at least positive semidefinite value of the simplex method assumption in practice many... Fractional power series neural network for solving a class of fractional optimal control problems with inequality constraints related to state! Slack variables precisely all equality constraints hierarchical IK algorithm was proposed that claims to avoid the instability.
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