polynomial 多项式
numerical optimization 数值优化
allocate 分配
resort to 采取
restore to 恢复到
capable 可行的 incapable 无法
simultaneous 同时
agile and maneuverable 敏捷与可操控性
continuous-time
discrete-time
quadrotor 四旋翼
state space representation 状态空间表示
differentially-flat
inherently 固有的
infinitesimal 无穷小
suboptimal 次优的
time-discretized 时间离散化
search and sampling-based 基于搜索和采样的
appoximation 近似 intracable 棘手的
system dynamics and input boundaries 系统动力学和输入边界
cost function 成本函数 complementary progress constraint 互补进度约束
intuitively 直观地讲
restricted to 受限于
maneuver 机动
specify 指定
aforementioned 前述的
traverse-dynamics 横向动力学
arc length 弧长
reference 参考
whereas 而
saturation 饱和
actuator 执行器
euler angle 欧拉角
refine 完善
feasible 可行的
take inspiration from 从...中获取灵感
non-convex 非凸 non-convexity 非凸性
interception 拦截
methodogy 方法论
respectively 分别
complement 补充 implement 实施
loss of generality 丧失一般性
fixed 固定的
coincide with 与...一致
decompose into 分解为
torque 扭矩
lap 一圈
substantial 重大的
ramp up 上升
thrust-to-weight ratio(TWR) 推力比
- Early works : Using polynomial trajectory formulations
- Recently works : **Using numerical optimization , but require waypoints to be allocated as cost or constraints at specific discrete times **
- This article propose : **propose a solution to the time allocation problem, by introducing a formulation of progress along the trajectory, which enable the simultaneous optimization of the time-allocation and the trajectory **
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continuous-time polynomials
- week-point : These polynomials are inherently smooth and therefore cannot represent rapid state or input changes at reasonable order, They cannot exploit the full actuator potential
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discrete-time state space representations
- methods :
- search and sampling-based methods
- optimization-based methods
- shortcoming : when faced the multiple waypoints, the time spent between any two waypoints is unknown, so this is ineffective for time-optimal trajectory generation through multiple waypoints
- methods :
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This paper's approach : formulate a progress measure for each waypoint along the trajectory. And introduce a (Complementary Progress Constraint)CPC, this allows completion only in proximity to a waypoint.
They formulate two factors which must complement each other
- One factor : the completion of a waypoint
- Another factor : local proximity to a waypoint
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General trajectory Optimization $$ x^*=arg minL(x) $$
$$ subject\ to: g(x)=0\ and\ h(x) \leq 0 $$ -
Multiple Shooting Method : 多重射击法
Represent a dynamic system in the state space
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system state(node) : xk
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actuation input : uk
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system evolution $$ \dot x = f_{dyn}(x,u) $$
$$ x_{k+1}-x_k-dt*f_{RK4}(x_k,u_k)=0 $$
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Time-Optimal Trajectory Optimization
Optimizing for a time-optimal trajectory : The only cost term : overall trajectory time $$ L(x)=t_N $$
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Passing Waypoints through Optimization
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Progress Measure Variables
To describe the progress throughout a track a measure that satisfy three requirements:
- Start at a defined value
- Reach a different value by the end of the trajectory
- Only change when a waypoint is passed within a certain tolerance
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Complementary Progress Constraints
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Tolerance Relaxation
Why this tolerance should be admitted?
- It's impractical to force trajectory to pass exactly through a waypoint
- negatively impacts the convergence behavior and time-optimality
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Optimization Problem Summary
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Quadrotor Dynamics
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- Velocity : The velocity distribution is rather similar for the human and time-optimal
- Acceleration ： The human varies considerably more
- didn't account for any latency correction of the whole pipeline
- In high speed , the aerodynamics model would be inaccurate
- The flight control Betaflight has more reliable system to human than time-optimal and the control strategies are not translate to accurate closed-loop control or desirable control-loop shaping