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Control Systems and Their Limitations

In this section, we step from kinematics to control. We first show how to reason with velocities (differential inverse kinematics, diff-IK), then close the loop with feedback, and finally summarize where classical pipelines struggle in practice.

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Differential Kinematics: A Smarter Approach

Instead of solving for joint positions directly, we can work with velocities:

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The Key Insight

If we know the relationship between joint velocities and end-effector velocities, we can control motion more smoothly: p˙=J(q)q˙\dot{p} = J(q) \dot{q}p˙​=J(q)q˙​

Where J(q)J(q)J(q) is the Jacobian matrix - the relationship between joint and task space velocities.

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Differential IK Solution

Given a desired end-effector velocity $\dot{p}^*$, find joint velocities: q˙=J(q)+p˙∗\dot{q} = J(q)^+ \dot{p}^*q˙​=J(q)+p˙​∗

Where J(q)+J(q)^+J(q)+ is the pseudo-inverse of the Jacobian.

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Adding Feedback Control

Open-loop tracking is brittle under modeling errors and disturbances. We close the loop by feeding back the tracking error.

Dealing with moving obstacles requires feedback control.

Real environments are dynamic and uncertain. We need feedback to handle:

- Modeling errors - Our equations aren’t perfect
- Disturbances - Unexpected forces or obstacles
- Sensor noise - Measurements have uncertainty

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Feedback Control Solution

Combine desired motion with error correction: q˙=J(q)+(p˙∗+k p Δ p)\dot{q} = J(q)^+ (\dot{p}^* + k_p \Delta p)q˙​=J(q)+(p˙​∗+k p​Δ p)

Where Δ p=p∗−p(q)\Delta p = p^* - p(q)Δ p=p∗−p(q) is the position error.

> Start with small k p k_p k p​ and increase gradually while monitoring oscillations. Use a watchdog (safety stop) and saturate commands to keep the system within safe limits.

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Why Classical Approaches Struggle

With differential reasoning and feedback, many tracking tasks are solvable—on paper. In practice, the system still breaks under real-world complexity for the reasons below.

Four key limitations of dynamics-based robotics approaches.

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1. Integration Challenges

Classical pipelines are built from separate modules:

- Sensing → State Estimation → Planning → Control → Actuation

Problems:

- Errors compound through the pipeline
- Brittle when any component fails
- Hard to adapt to new tasks or robots

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2. Limited Scalability

Traditional methods struggle with:

- High-dimensional sensor data (cameras, LIDAR)
- Multi-task scenarios (each task needs custom planning)
- Multi-modal integration (vision + touch + proprioception)

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3. Modeling Limitations

Real-world physics is complex:

- Contact dynamics - Hard to model precisely
- Deformable objects - Beyond rigid-body assumptions
- Friction and compliance - Difficult to characterize

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4. Ignoring Data Trends

Classical methods don’t leverage:

- Growing robotics datasets - Millions of demonstrations available
- Cross-robot learning - Insights from other platforms
- Community knowledge - Decentralized data collection

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The Learning Alternative

To address these limitations, we contrast a classical modular pipeline with an end-to-end learning policy.

classical
learning

Classical Robotics Approach:

    Perception → State Estimation → Planning → Control → Actuation

Challenges:

- Each module needs expert tuning
- Errors compound through pipeline
- Hard to adapt to new tasks/robots
- Requires precise world models

This is the promise of robot learning!

> The Best of Both Worlds: Modern robot learning often combines classical insights with learning. For example one can combine learning with safety constraints from control theory

> Pure learning vs pure classical is a false dichotomy - hybrid approaches have had their successes

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Key Takeaways

- Classical robotics relies on explicit mathematical models and expert knowledge
- Forward kinematics is straightforward, but is viable only in quite simple scenarios. Inverse kinematics is more general, but it can be challenging to develop in practice
- Differential kinematics works with velocities rather than positions for better control
- Classical approaches struggle with integration, scalability, modeling accuracy, and data utilization
- Learning-based methods offer solutions to these fundamental limitations
- The future lies in hybrid approaches that combine classical insights with learning capabilities

> Up next, we’ll show how learning-based methods (reinforcement learning and imitation learning) absorb some of this complexity by optimizing directly from data.

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References

For a full list of references, check out the tutorial.

- Feedback Systems: An Introduction for Scientists and Engineers (2008)
  Karl Johan Åström and Richard M. Murray
  A comprehensive introduction to feedback control systems, covering the principles that underlie closed-loop control in robotics.
  Book Website

- Real-Time Obstacle Avoidance for Manipulators and Mobile Robots (1986)
  Oussama Khatib
  A seminal paper introducing the artificial potential field method for obstacle avoidance, demonstrating how feedback can be used for reactive control in dynamic environments.
  DOI:10.1177/027836498600500106

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