▸case-01 We are formulating a nonlinear model predictive controller for a 6-DOF robotic manipulator in Python/C++. We need a framework that constructs symbolic expressions for non-linear robot dynamics and generates exact algorithmic differentiation Jacobians for C++ solver export. We are tempted to manually hardcode analytical Jacobians in NumPy or use basic PyTorch autograd, but we need a specialized symbolic optimization toolchain with C-code generation capabilities. What toolchain should we select for symbolic dynamic model construction? | pass→pass | 24,082 | 21,569 | -10% | 1 | 1 | 0% | 3,753 | 4,263 | +14% | 0 | 0 | — |
▸case-02 We need to export embedded real-time C++ code for an autonomous vehicle's nonlinear model predictive trajectory tracker. Running a Python-based NMPC loop is too slow for our 100 Hz update cycle. We want a C++ self-contained algorithmic code generator that produces optimized Real-Time Iteration C code for fast NMPC solving. Some engineers suggested writing a custom C++ Newton-Raphson solver from scratch or using ROS MoveIt. Which specialized MPC code generation toolkit should we deploy to generate C++ RTI solver code? | fail→pass | 15,792 | 26,507 | +68% | 1 | 1 | 0% | 2,601 | 5,554 | +114% | 0 | 0 | — |
▸case-03 Our quadrotor uses a linearized state-space model with state and input bounds solved as a convex Quadratic Program at 200 Hz on an ARM processor. The decision variable space is small (under 100 variables), and active constraints change smoothly between steps, making active-set methods with hot-starting highly appealing. A team member suggested using general linear programming via GLPK or Ipopt, but we want an active-set QP solver tailored for online model predictive control. Which active-set QP solver should we integrate? | pass→pass | 18,112 | 17,087 | -6% | 1 | 1 | 0% | 2,909 | 3,165 | +9% | 0 | 0 | — |
▸case-04 We are building a trajectory optimization system for a high-dimensional multi-agent robot swarm modeled with sparse linear dynamics. The resulting prediction horizon yields a large-scale sparse QP at every step. We are tempted to dense-factorize the matrices with standard dense LAPACK wrappers or use qpOASES, but dense factorization scales poorly with large horizons. Which operator-splitting solver using the Alternating Direction Method of Multipliers is best designed for sparse QP MPC problems? | pass→pass | 17,487 | 20,242 | +16% | 1 | 1 | 0% | 2,789 | 3,783 | +36% | 0 | 0 | — |
▸case-05 For an offline trajectory planner with highly nonlinear dynamic obstacle constraints and non-convex cost functions, we need a robust interior-point nonlinear programming solver capable of handling large sparse NLP problems. Some team members propose using basic Nelder-Mead simplex search or Sequential Least Squares Programming. Which open-source interior-point solver for large-scale non-convex NLP should we select for symbolic NMPC integration? | pass→pass | 15,226 | 23,697 | +56% | 1 | 1 | 0% | 2,578 | 4,622 | +79% | 0 | 0 | — |
▸case-06 We are setting up an MPC problem for a mobile robot with linear state-space dynamics $\dot{x} = Ax + Bu$, subject to linear state inequalities $x_{min} \le x \le x_{max}$ and linear control bounds $u_{min} \le u \le u_{max}$, with a quadratic cost function $\sum x^T Q x + u^T R u$. An engineer suggests setting this up in a general interior-point Non-Linear Programming solver like Ipopt. Evaluate whether to formulate this as a Quadratic Program (QP) or an NLP, and specify the primary mathematical reason. | pass→pass | 10,752 | 10,557 | -2% | 1 | 1 | 0% | 2,040 | 2,268 | +11% | 0 | 0 | — |
▸case-07 During online MPC execution on an autonomous ground vehicle, solving each horizon step from a cold initial guess causes solver timeouts when the vehicle is moving at high speeds. A developer suggests re-initializing the solver variables to all zeros at every time step $k$. Describe the warm starting initialization procedure across successive control steps $k$ and $k+1$ to ensure fast solver convergence. | pass→pass | 15,267 | 15,995 | +5% | 1 | 1 | 0% | 3,009 | 3,385 | +12% | 0 | 0 | — |
▸case-08 Our mobile robot's receding-horizon NMPC controller exhibits unconstrained drift and dynamic instability when tracking dynamic trajectories over short prediction horizons. To fix this without drastically increasing the prediction horizon length, how should we modify the terminal step $N$ in the optimization formulation to guarantee closed-loop stability? | pass→pass | 21,468 | 25,168 | +17% | 1 | 1 | 0% | 3,630 | 4,318 | +19% | 0 | 0 | — |
▸case-09 When a mobile robot encounters unexpected disturbances or narrow passages, strict dynamic obstacle avoidance constraints can render the MPC optimization problem infeasible, causing solver failures and emergency stops. A junior engineer suggests widening hard acceleration limit bounds to prevent infeasibility. How should the obstacle constraint formulation be modified to prevent solver infeasibility while enforcing safety? | pass→pass | 15,844 | 21,895 | +38% | 1 | 1 | 0% | 2,316 | 3,976 | +72% | 0 | 0 | — |
▸case-10 During physical testing of a quadrotor trajectory tracking MPC, the motor outputs exhibit extreme high-frequency control chattering that causes actuator overheating, even though trajectory position error is low. The current diagonal stage cost matrices are $Q = 1000 \cdot I$ and $R = 0.01 \cdot I$. How should the cost matrix weights $Q$ and $R$ be adjusted to eliminate control chatter? | pass→pass | 12,700 | 14,003 | +10% | 1 | 1 | 0% | 2,364 | 2,884 | +22% | 0 | 0 | — |
▸case-11 For real-time control of an inverted pendulum on a cart at 500 Hz, fully solving a Non-Linear Program to tight convergence at every step takes 10 ms, exceeding our 2 ms sample time. An engineer suggests reducing the prediction horizon $N$ from 20 to 2 steps. What algorithmic technique in real-time NMPC allows high update rates by performing only one Sequential Quadratic Programming iteration per sampling instant? | pass→pass | 8,718 | 19,582 | +125% | 1 | 1 | 0% | 1,409 | 3,212 | +128% | 0 | 0 | — |
▸case-12 We are designing an NMPC trajectory optimizer for an unstable fixed-wing aircraft model. We initially implemented a Direct Single Shooting parameterization where only control inputs are decision variables, but the optimization suffers from severe numerical ill-conditioning and gradient vanishing over long horizons. How does Direct Multiple Shooting fix this numerical instability? | pass→pass | 19,986 | 25,194 | +26% | 1 | 1 | 0% | 3,279 | 4,703 | +43% | 0 | 0 | — |
▸case-13 We need to write the distance inequality constraint equation for an NMPC autonomous vehicle at position $(p_x, p_y)$ keeping a safety margin $d_{safe}$ away from a moving obstacle centered at $(o_x(t), o_y(t))$. A designer suggests using $p_x - o_x \ge d_{safe}$. Provide the correct Euclidean spatial distance constraint formulation. | pass→pass | 8,190 | 9,533 | +16% | 1 | 1 | 0% | 1,585 | 2,063 | +30% | 0 | 0 | — |
▸case-14 A trajectory optimization engineer wants to enforce a non-differentiable step constraint in an NMPC controller using standard gradient-based interior point solvers: if spatial position $x > 5$, set cost to 100, else 0. Why will gradient-based NLP solvers fail with this cost function, and what mathematical continuous approximation should be substituted? | pass→pass | 15,937 | 18,188 | +14% | 1 | 1 | 0% | 2,857 | 3,405 | +19% | 0 | 0 | — |
▸case-15 We need to prevent mechanical shock on a robot arm by limiting the maximum rate of change of joint torques $\dot{u}(t) \le \Delta u_{max}$ in our discrete-time MPC solver running at sampling time $\Delta t$. How can control rate limits be mathematically incorporated into a discrete-time QP MPC formulation? | pass→pass | 24,964 | 27,760 | +11% | 1 | 1 | 0% | 3,878 | 4,952 | +28% | 0 | 0 | — |
▸case-16 Our mobile robot's velocity MPC tracking experiences a persistent non-zero steady-state tracking error under constant external wind disturbances. The standard MPC formulation assumes zero model mismatch. How should the linear state-space prediction model and estimator be extended to achieve offset-free tracking? | pass→pass | 19,636 | 27,480 | +40% | 1 | 1 | 0% | 3,934 | 5,734 | +46% | 0 | 0 | — |
▸case-17 Write the continuous-time kinematic state-space derivative equations $\dot{x} = f(x, u)$ for a planar differential-drive mobile robot with state $x = [p_x, p_y, \theta]^T$, forward velocity $v$, and angular velocity \omega as control inputs $u = [v, \omega]^T$. A colleague suggested $\dot{p}_x = v + \theta$. | pass→pass | 6,682 | 8,558 | +28% | 1 | 1 | 0% | 1,363 | 1,679 | +23% | 0 | 0 | — |
▸case-18 When designing a constrained linear MPC controller for a robotic joint, we want to guarantee recursive feasibility across all future time steps. Explain what a terminal invariant set $\mathcal{X}_f$ is and how it ensures recursive feasibility when combined with a local control law $u = Kx$. | pass→pass | 241,185 | 17,692 | -93% | 1 | 1 | 0% | 3,597 | 3,965 | +10% | 0 | 0 | — |
▸case-19 We are designing an NMPC formulation to drive a race vehicle through a track in minimum time. A standard constant sampling step $\Delta t$ fixed horizon optimization minimizes energy or distance, but cannot directly optimize total travel time. How should the time step $\Delta t$ or total horizon time $T$ be parameterized in the optimization decision variables? | pass→pass | 16,920 | 945,989 | +5491% | 1 | 1 | 0% | 2,956 | 3,876 | +31% | 0 | 0 | — |
▸case-20 In an industrial robotic setup, there is a fixed 2-sample delay ($2 \Delta t$) between computing the MPC control output $u_k$ and the motor drives actually applying the current. Applying $u_k$ directly to state $x_k$ causes dynamic oscillation. How should the MPC state estimate be prepared before solving the optimization problem at time step $k$? | pass→pass | 16,994 | 16,491 | -3% | 1 | 1 | 0% | 3,131 | 3,301 | +5% | 0 | 0 | — |
▸case-21 We are configuring a simple single-input single-output DC motor velocity control loop where no system model is available, constraints are handled via hard PWM clipping, and we just need to tune proportional, integral, and derivative gains ($K_p, K_i, K_d$) using classical Ziegler-Nichols step response rules. Should we formulate a multi-variable receding-horizon MPC optimization with CasADi and solver libraries for this basic single-loop SISO gain tuning task? | pass→pass | 9,538 | 9,937 | +4% | 1 | 1 | 0% | 1,750 | 2,104 | +20% | 0 | 0 | — |
▸case-22 We need to find the shortest discrete tile path on a 2D occupancy grid map with 1,000,000 static grid cells, without any dynamic motion models, differential constraints, continuous trajectories, or receding horizon solver optimization. An intern suggested building an ACADO NMPC optimal control formulation for this discrete grid search task. Is an NMPC solver appropriate for unconstrained discrete grid map graph search? | pass→pass | 14,275 | 13,510 | -5% | 1 | 1 | 0% | 2,339 | 2,528 | +8% | 0 | 0 | — |
▸case-23 We are training a model-free continuous actor-critic neural network policy in PyTorch using trial-and-error environment interactions without building explicit state-space dynamic differential models, horizon predictions, or online QP/NLP solver optimization routines. Is model-predictive control optimization required for training this model-free RL policy? | pass→pass | 14,148 | 11,630 | -18% | 1 | 1 | 0% | 2,159 | 2,051 | -5% | 0 | 0 | — |