Let's break this rocket control problem down into its fundamental components:
1. **State variables**:
- Position (x)
- Velocity (v)
2. **Control variable**:
- Acceleration (a)
3. **Discrete time model**:
- x(t+1) = x(t) + v(t)
- v(t+1) = v(t) + a(t)
4. **Constraints**:
- Initial: x(0) = 0, v(0) = 0
- Final: x(T) = 1, v(T) = 0
- T = total time units
5. **Optimization objectives**:
- Minimize total fuel: sum(|a(t)|) from t=0 to T-1
- OR minimize max thrust: max(|a(t)|) for all t
Key insights:
1. **Symmetry**: The optimal solution is likely symmetrical, with acceleration in the first half and deceleration in the second half.
2. **Fuel vs. thrust tradeoff**: Minimizing total fuel may lead to higher peak thrust, while minimizing max thrust may consume more total fuel.
3. **Time-fuel tradeoff**: A longer time T allows for lower accelerations but may consume more total fuel due to fighting gravity longer.
4. **Discretization effects**: The discrete time model is an approximation. Finer time steps (smaller than 1) would yield a more accurate solution.
5. **Linear programming formulation**: This allows for efficient solving, but may not capture all real-world complexities.
Elon Musk-inspired approach:
1. **Question assumptions**: Is a linear model sufficient? Should we consider gravity, air resistance, or variable mass?
2. **Iterate and simulate**: Develop a simple simulation to test various control strategies before optimizing.
3. **Consider reusability**: How would the problem change if we needed to reserve fuel for landing?
4. **Look for analogues**: Are there similar problems in other fields (e.g. finance, logistics) we can learn from?
5. **Think bigger**: How does this problem scale to interplanetary missions or reusable rockets?
By breaking down the problem and questioning assumptions, we can develop more innovative and robust solutions, potentially revolutionizing rocket control strategies.
Rocket
Mini tools
Open94418Released 1y ago100% FreeLet's break this rocket control problem down into its fundamental components: 1. **State variables**: - Position (x) - Velocity (v) 2. **Control variable**: - Acceleration (a) 3. **Discrete time model**: - x(t+1) = x(t) + v(t) - v(t+1) = v(t) + a(t) 4. **Constraints**: - Initial: x(0) = 0, v(0) = 0 - Final: x(T) = 1, v(T) = 0 - T = total time units 5. **Optimization objectives**: - Minimize total fuel: sum(|a(t)|) from t=0 to T-1 - OR minimize max thrust: max(|a(t)|) for all t Key insights: 1. **Symmetry**: The optimal solution is likely symmetrical, with acceleration in the first half and deceleration in the second half. 2. **Fuel vs. thrust tradeoff**: Minimizing total fuel may lead to higher peak thrust, while minimizing max thrust may consume more total fuel. 3. **Time-fuel tradeoff**: A longer time T allows for lower accelerations but may consume more total fuel due to fighting gravity longer. 4. **Discretization effects**: The discrete time model is an approximation. Finer time steps (smaller than 1) would yield a more accurate solution. 5. **Linear programming formulation**: This allows for efficient solving, but may not capture all real-world complexities. Elon Musk-inspired approach: 1. **Question assumptions**: Is a linear model sufficient? Should we consider gravity, air resistance, or variable mass? 2. **Iterate and simulate**: Develop a simple simulation to test various control strategies before optimizing. 3. **Consider reusability**: How would the problem change if we needed to reserve fuel for landing? 4. **Look for analogues**: Are there similar problems in other fields (e.g. finance, logistics) we can learn from? 5. **Think bigger**: How does this problem scale to interplanetary missions or reusable rockets? By breaking down the problem and questioning assumptions, we can develop more innovative and robust solutions, potentially revolutionizing rocket control strategies.
