Stack of Tasks (SoT) Control Strategy

• Stack of Tasks (SoT) is a hierarchical control strategy that solves cascaded quadratic programs to enforce prioritized task objectives under both equality and inequality constraints.
• It leverages null-space projections and hierarchical quadratic programming to ensure that safety-critical tasks are met before secondary objectives, maintaining strict lexicographic precedence.
• Recent formulations integrate set invariance via control barrier functions and allow time-varying priorities for smooth task insertion, switching, and multi-robot coordination.

Stack of Tasks (SoT) is a hierarchical, local control strategy that solves a cascade of Quadratic Programs at each control cycle to satisfy prioritized goals in error space under equality and inequality constraints. In its classical form, higher-priority tasks must be satisfied first, and lower-priority tasks are executed only within the null space of the tasks above them so they do not interfere. More recent formulations generalize SoT from pointwise tracking to set invariance and attractivity, encode tasks through Control Barrier Functions (CBFs) or Control Lyapunov Functions (CLFs), and treat time-varying priorities, task insertion, learned value functions, and supervisory switching within a unified constrained-optimization framework (Domínguez et al., 2022, Notomista et al., 2020).

1. Canonical definition and differential structure

SoT is grounded in differential kinematics for redundant robots. A task is defined by a smooth mapping such as $x=f(q)$ or $\sigma=k(q)$, with Jacobian $J(q)=\partial f/\partial q$ or $J(q)=\partial k/\partial q$, so that

$$\dot{x} = J(q)\dot{q}, \qquad \dot{\sigma}=J(q)\dot{q}.$$

Task regulation is typically written in terms of an error, for example $e_i=x_i^\ast-x_i(q)$, together with a proportional desired task-space evolution such as $\dot{e}^\ast=-K_p e$ or $\dot{x}_i^\ast=\Omega_i e_i$ (Notomista et al., 2023, Domínguez et al., 2022, Katyara et al., 2021).

The defining structural feature of SoT is strict priority. In the null-space formulation, a primary task is solved first, and secondary tasks are projected into the null space of higher-priority Jacobians. For two tasks, one canonical form is

$$\dot{q}_d = J_0^\dagger \Omega_0 e_0 + N_0(q) J_1^\dagger \Omega_1 e_1, \qquad N_0(q)=I-J_0^\dagger J_0.$$

For multiple tasks, the hierarchy is extended recursively through null-space projectors or equivalent prioritized least-squares constructions, so that each lower-priority command is restricted to directions that do not alter the achieved higher-priority behavior (Katyara et al., 2021, Notomista et al., 2023, Adami et al., 14 Aug 2025).

This structure is central to redundancy resolution. A robot is redundant when it has more degrees of freedom than needed for a single task, and SoT exploits that redundancy to handle several objectives simultaneously, including tracking, obstacle avoidance, manipulability, and joint-limit avoidance. In this sense, SoT is not a single controller but a control architecture for organizing multiple objectives by precedence (Adami et al., 14 Aug 2025).

2. Classical hierarchical QP formulations

A second major realization of SoT is the hierarchical quadratic programming (HQP) approach. Here tasks are expressed as equality or inequality constraints in error space and solved as a sequence of QPs, one level per priority. In the inequality form reported for a single task,

$$\boldsymbol{J}\dot{\boldsymbol{q}} \leq \dot{\boldsymbol{e}}^\ast(\boldsymbol{q}),$$

and infeasibility is handled through slack variables by minimizing the slack magnitude subject to relaxed inequalities. For a hierarchy with priority levels $\sigma=k(q)$0, tasks at each level are stacked into a bound form $\sigma=k(q)$1, and higher-priority slack solutions are frozen as “priority consistency constraints” when solving lower levels (Domínguez et al., 2022).

This classical HQP view emphasizes local reactivity. Because SoT computes instantaneous control actions at high frequency based on the current robot and environment state, without modeling temporal state evolution over horizons, it is well-suited to handling local disturbances online. At the same time, this same locality limits its ability to address goals that require non-quadratic objectives or locally suboptimal actions to avoid local minima. The BT-integrated formulation explicitly identifies these limitations and motivates higher-level task switching as a complement to low-level prioritized control (Domínguez et al., 2022).

Classical SoT and HQP also differ in how they treat priorities. Null-space projection imposes hard lexicographic precedence through projector algebra, whereas HQP imposes it by solving higher-priority levels first or via prioritized optimization. Both preserve the core SoT principle that lower-priority objectives must not alter higher-priority ones, but both can suffer discontinuities when task stacks switch (Notomista et al., 2023).

3. Set-based and extended set-based tasks

A major reformulation of SoT replaces pointwise task satisfaction by set invariance and attractivity. In the extended set-based task framework, a task is specified by

$\sigma=k(q)$2

where $\sigma=k(q)$3 is continuously differentiable. The execution objective is to render $\sigma=k(q)$4 forward invariant and asymptotically stable. This extends classical set-based tasks such as joint limits and obstacle margins to include “going towards a set” in addition to “staying in a set,” thereby unifying regulation or tracking tasks with safety tasks (Notomista et al., 2020, Notomista et al., 2023).

Tracking appears as a special case. With desired trajectory $\sigma=k(q)$5, choosing

$\sigma=k(q)$6

yields the zero superlevel set $\sigma=k(q)$7. Stabilizing and keeping this set invariant recovers the original tracking or regulation objective (Notomista et al., 2020, Notomista et al., 2023).

The CBF encoding is defined on control-affine dynamics

$\sigma=k(q)$8

An output time-varying CBF satisfies

$\sigma=k(q)$9

with $J(q)=\partial f/\partial q$0 an extended class-$J(q)=\partial f/\partial q$1 function. Any Lipschitz controller satisfying the corresponding affine inequality renders $J(q)=\partial f/\partial q$2 forward invariant and asymptotically stable. In manipulator kinematics, these constraints reduce to affine half-spaces on joint velocities; in dynamic models, relative-degree issues are addressed by auxiliary or higher-order CBFs such as $J(q)=\partial f/\partial q$3 (Notomista et al., 2020, Notomista et al., 2023).

The optimization step executes multiple tasks through a single convex QP. In one form,

$J(q)=\partial f/\partial q$4

subject to one relaxed CBF inequality per task and linear priority relations on the slack vector. Safety-critical tasks are enforced as hard constraints by fixing the corresponding slack to zero, whereas non-safety tasks are prioritized softly through pairwise slack inequalities such as $J(q)=\partial f/\partial q$5 with $J(q)=\partial f/\partial q$6 (Notomista et al., 2020). A related CLF-based formulation uses learned or prescribed value functions $J(q)=\partial f/\partial q$7 and solves a single convex QP with CLF decrease constraints and linear priority inequalities on the slack vector $J(q)=\partial f/\partial q$8, thereby extending SoT to tasks encoded by reinforcement-learning value functions and to coordinated multi-robot behavior (Notomista, 2023).

This suggests a broad reinterpretation of SoT: rather than a hierarchy of projector equations or sequential QPs alone, it becomes a family of constrained-optimization schemes in which task semantics are expressed as set invariance, attractivity, or value-function decrease.

4. Time-varying priorities, switching, and insertion

A persistent issue in classical SoT is discontinuity during task-stack changes. The extended set-based formulations address this directly. When priorities are encoded by a time-varying matrix $J(q)=\partial f/\partial q$9 and $J(q)=\partial k/\partial q$0 is Lipschitz in time, the optimal controller $J(q)=\partial k/\partial q$1 of the single-QP formulation is Lipschitz continuous, which avoids discontinuities when priorities change (Notomista et al., 2020).

Task insertion and removal are handled by smooth scheduling. A new task can be introduced through a continuous ramp $J(q)=\partial k/\partial q$2 that is identically zero before the insertion time and identically one after a finite transition interval. The inserted task constraint is then multiplied by $J(q)=\partial k/\partial q$3, ensuring continuity of the optimal control during insertion; analogous scheduling with $J(q)=\partial k/\partial q$4 gives continuity during removal (Notomista et al., 2020).

A related mechanism appears in the 2023 ESB framework through either relaxed priority constraints,

$J(q)=\partial k/\partial q$5

or input blending,

$J(q)=\partial k/\partial q$6

over a finite interval. Relaxation yields an automatically prioritized, numerically well-posed stack, while blending preserves continuity and unchanged relative priorities during stack transitions (Notomista et al., 2023).

Compared with null-space SoT and HQP, these approaches shift the emphasis from exact lexicographic enforcement to continuity, feasibility, and native handling of inequality or set-based tasks. The trade-off is explicit in the source material: strict lexicographic behavior is approximated via weights, slack relations, or large $J(q)=\partial k/\partial q$7, rather than enforced exactly except where slacks are fixed to zero for safety (Notomista et al., 2020, Notomista et al., 2023).

5. Representative systems and empirical realizations

The literature represented here spans manipulator control, collaborative manipulation, multi-robot coordination, and learned hierarchy design.

Paper	Setting	Main SoT contribution
(Notomista et al., 2020)	7-DoF redundant manipulator in simulation	Single-QP CBF execution, hard safety, soft priorities, continuous switching
(Notomista et al., 2023)	Kinematic and dynamic manipulators, simulations and KUKA LBR iiwa 7 R800	Formal ESB task definition, input bounds, quantitative comparison to HQP
(Domínguez et al., 2022)	Franka Emika Panda with Kinect V2	BT-configured HQP SoT for reactive task switching
(Katyara et al., 2021)	Collaborative fine manipulation with visuo-tactile sensing	Intuitive SoT with posture-driven switching and force-based manipulation
(Notomista, 2023)	Team of 6 planar robots in simulation	CLF-QP execution of prioritized learned multi-robot tasks
(Adami et al., 14 Aug 2025)	ABB mobile-YuMi in simulation and hardware	Learning SoT hierarchy and parameters by RL plus Genetic Programming

In the redundant manipulator case study, a 7-DoF arm with a camera on the end-effector executed joint-limit avoidance, visual servoing, and end-effector positioning. The safety-critical joint-limit task remained at highest priority with $J(q)=\partial k/\partial q$8; visual servoing was inserted at $J(q)=\partial k/\partial q$9 s, position at $$\dot{x} = J(q)\dot{q}, \qquad \dot{\sigma}=J(q)\dot{q}.$$0 s, and the relative order of the latter two was swapped at $$\dot{x} = J(q)\dot{q}, \qquad \dot{\sigma}=J(q)\dot{q}.$$1 s. Reported plots showed smooth joint velocities, positive joint-limit CBF values, and performance CBFs driven toward zero as priorities allowed (Notomista et al., 2020).

The extended 2023 framework validated ESB tasks in kinematic simulations with a 3-link planar manipulator, in dynamic simulations with torque bounds, and on a KUKA LBR iiwa 7 R800. Reported behaviors included smooth switching among dependent end-effector tasks, task insertion and removal with continuity preservation, nonnegative joint-limit CBFs, and bounded torques through integral CBFs (Notomista et al., 2023).

The BT-plus-SoT framework assigns supervisory logic to a Behavior Tree running at lower frequency and control to an SoT solver running at higher frequency. The BT tick frequency was reported as 2–110 Hz and the SoT solve frequency as 220–1500 Hz. On a Franka Emika Panda performing a pick-place-push task with a 40 mm cube, the reported robustness over 50 trials across 5 start positions was 94% overall success, 90% first-attempt success, and 100% second-attempt success after one automatic recovery. In disturbance experiments, local disturbances yielded 92% success over 25 trials, while global disturbances yielded 88% success over 25 trials (Domínguez et al., 2022).

The intuitive SoT formulation for collaborative fine manipulation couples hard-priority Cartesian positioning and force tasks with soft-priority manipulability and joint-limit avoidance. It uses a traded visuo-tactile sensing topology with a tracking camera at 5 Hz, an eye-in-hand detection camera at 25 Hz, tactile sensing at 115.2 kHz, and a 1 kHz control loop. In the reported assembly and disassembly tasks, human-robot collaboration achieved task repeatability values of 0.9142 and 0.9607, cumulative posture deviation values of 0.1169% and 0.0993%, and grasp correction values of 1.955 mm and 1.181 mm for the two tasks, while task coordination latency remained substantially larger than in human-human collaboration (Katyara et al., 2021).

Multi-robot and learned-task variants further broaden the SoT scope. A team of 6 planar single-integrator robots executed a hexagonal formation task and three go-to-goal tasks with priorities changed online, and the single-QP CLF framework preserved feasibility through slack while respecting relative priorities (Notomista, 2023). In the hierarchy-learning approach, a mobile-YuMi with two 7-DoF arms and a 3-DoF base learned priority orders and task parameters from user-defined costs. Reported learned stacks commonly prioritized obstacle avoidance first, followed by inverse kinematics, manipulability maximization, and distance from mechanical joint limits, with positions 3 and 4 able to swap depending on the cost weights (Adami et al., 14 Aug 2025).

6. Limitations, trade-offs, and research directions

Several limitations recur across formulations. Classical SoT is local and reactive, but it does not model temporal state evolution and can struggle with non-quadratic objectives or situations requiring locally suboptimal actions (Domínguez et al., 2022). Null-space and HQP methods can induce discontinuities during stack switches, particularly when tasks are swapped without smoothing (Notomista et al., 2023, Notomista et al., 2020).

Set-based and optimization-based variants introduce their own assumptions. The ESB-CBF formulations assume accurate models, smooth outputs, Lipschitz $$\dot{x} = J(q)\dot{q}, \qquad \dot{\sigma}=J(q)\dot{q}.$$2 and $$\dot{x} = J(q)\dot{q}, \qquad \dot{\sigma}=J(q)\dot{q}.$$3, and continuously differentiable task functions $$\dot{x} = J(q)\dot{q}, \qquad \dot{\sigma}=J(q)\dot{q}.$$4. Safety-task feasibility requires adequate control authority, and overly tight or fast $$\dot{x} = J(q)\dot{q}, \qquad \dot{\sigma}=J(q)\dot{q}.$$5 gains can create conflicts. Soft prioritization may permit small violations of lower-priority tasks, and exact lexicographic priority is approximated through slack relations or weights rather than strictly enforced except for hard safety tasks (Notomista et al., 2020, Notomista et al., 2023).

Perception-driven and learned variants also inherit sensing and training limitations. The BT-plus-SoT experiments identify fiducial tracking errors as the main source of unrecoverable failures (Domínguez et al., 2022). The collaborative visuo-tactile formulation reports coordination latency due to heterogeneous sensing rates and notes SVM classification limits and empirically tuned slip thresholds (Katyara et al., 2021). The hierarchy-learning method reports slower obstacle-avoidance response on hardware because of sensor noise and delays, and the lack of a global navigation planner can cause the base to move farther from the target than necessary (Adami et al., 14 Aug 2025).

The cumulative picture is that SoT has evolved from a strict projector-based hierarchy for prioritized inverse kinematics into a broader class of hierarchical constrained-optimization methods. Within this evolution, the main axes of current development are explicit treatment of safety and set invariance, continuous re-prioritization, integration with supervisory or perceptual modules, execution of learned tasks, and automatic synthesis of the hierarchy itself (Notomista et al., 2020, Notomista et al., 2023, Notomista, 2023, Adami et al., 14 Aug 2025).