1 Introduction
1.1 Contributions
1.2 Related work
1.2.1 Open-loop MAVP trajectory planning
1.2.2 Closed-loop MAVP trajectory planning
1.2.3 Non-linear model predictive control and unified planning and control of MAV(P)s
1.3 Paper organisation
2 Preliminaries
2.1 Notation
2.2 Quadrotor with swung payload model
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rigid, massless cable with free suspension points,
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quadrotor centre of gravity, centroid, and cable suspension point coincide,
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no aerodynamic drag effects on the cable.
2.2.1 Quadrotor inputs
2.2.2 Aerodynamic drag effects
2.2.3 System kinematics and dynamics
2.2.4 Full MAVP model
Notation | Definition |
---|---|
\(m_q, m_l; g \in {\mathbb {R}}\) | Mass of quadrotor, load; Gravitational acceleration |
\(l; \theta _l, \phi _l \in {\mathbb {R}}\) | Cable length; Payload suspension angles |
\(\varvec{p}_q, \varvec{p}_l \in {\mathbb {R}}^3\) | Position of quadrotor, payload in \(\{I\}\) |
\(\varvec{q}, \varvec{{\dot{q}}} \in {\mathbb {R}}^5\) | MAVP configuration, and its time derivative |
\(\varvec{u} \in {\mathbb {R}}^3\) | Quadrotor input commands |
\(\varvec{F}, \varvec{F_u} \in {\mathbb {R}}^n\) | General, control input force in \(\{I\}\) |
\(\varvec{x}_c, \varvec{x}_q, \varvec{x} \in {\mathbb {R}}^n\) | Quadrotor input, system, and full MAVP model state |
2.3 Obstacle model
2.3.1 Obstacle ellipsoid definitions
2.3.2 Obstacle motion prediction
2.4 MAVP-obstacle collision avoidance requirements
2.4.1 Point to ellipsoid distance
2.4.2 Quadrotor and payload proximity
2.4.3 Rigid cable proximity
3 Online and closed-loop MAVP trajectory generation
3.1 Method overview
3.1.1 Receding horizon dynamic planning
3.1.2 Local trajectory generation
3.2 Costs
3.2.1 Point-to-point navigation
3.2.2 Potential field based obstacle separation
3.2.3 Input magnitude regulation
3.2.4 Payload suspension angles regulation
3.3 Constraints
3.3.1 MAVP dynamics
3.3.2 State and input limits
3.3.3 Collision-free planning
3.3.4 Workspace limits
3.3.5 Scalability to large obstacle rich workspaces
3.4 Optimisation algorithm
3.4.1 Costs
3.4.2 Constraints
3.5 Theoretical analysis
3.5.1 Problem dimensionality
3.5.2 Optimality and feasibility
4 System setup and framework
4.1 System properties and hardware
Quad. mass | 500 g | Quad. drag const. \(k_{Dq}\) | 0.28 |
Load mass | 11 g | Load drag const. \(k_{Dl}\) | 0.00177 |
Cable length | 0.77 m | Max. \(\left| {\bar{\theta }}_q\right| \), \(\left| {\bar{\phi }}_q\right| \) input | \(15^\circ \) |
Max. \(\left| {\bar{w}}_q\right| \) input | 1 m/s | Detection range | 3.5 m |
4.2 Workspace
4.3 Programmed control system framework
Navigation \(w_\text {nav}\) | 1.0 | Inputs \(w_\text {in}\) | 0.01 |
Potential field \(w_\text {pf}\) | 1.2 | Swing Angles \(w_\text {swing}\) | 0.001 |
Slacks \(w_\text {slack}\) | 10000 |
4.4 Cascaded Kalman filter state estimator
5 Simulation study
5.1 Scalability of the optimisation problem
5.1.1 Scaling with number of planning stages
5.1.2 Scaling with number of dynamic obstacles
Algorithm | Off-line (s) | Time-to-goal (s) | Total (s) |
---|---|---|---|
Simple task | |||
NMPC | N/A | 2.65 | 2.65 |
Pre-Generated | 2.91 | 2.25 | 5.16 |
Minimal Swing | N/A | 7.10 | 7.10 |
Difficult task | |||
NMPC | N/A | 5.35 | 5.35 |
Pre-Generated | 10.54 | 4.85 | 15.39 |
Minimal Swing | N/A | 18.55 | 18.55 |