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67 lines
2.1 KiB
Markdown
67 lines
2.1 KiB
Markdown
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# Arm OLS with angle offset
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If the arm encoder doesn't read zero degrees when the arm is horizontal, the fit
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for `Kg` will be wrong. An angle offset should be added to the model like so.
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```
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dx/dt = -Kv/Ka x + 1/Ka u - Ks/Ka sgn(x) - Kg/Ka cos(angle + offset)
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```
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Use a trig identity to split the cosine into two terms.
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```
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dx/dt = -Kv/Ka x + 1/Ka u - Ks/Ka sgn(x) - Kg/Ka (cos(angle) cos(offset) - sin(angle) sin(offset))
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dx/dt = -Kv/Ka x + 1/Ka u - Ks/Ka sgn(x) - Kg/Ka cos(angle) cos(offset) + Kg/Ka sin(angle) sin(offset)
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```
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Reorder multiplicands so the offset trig is absorbed by the OLS terms.
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```
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dx/dt = -Kv/Ka x + 1/Ka u - Ks/Ka sgn(x) - Kg/Ka cos(offset) cos(angle) + Kg/Ka sin(offset) sin(angle)
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```
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## OLS
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Let `α = -Kv/Ka`, `β = 1/Ka`, `γ = -Ks/Ka`, `δ = -Kg/Ka cos(offset)`, and `ε = Kg/Ka sin(offset)`.
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```
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dx/dt = αx + βu + γ sgn(x) + δ cos(angle) + ε sin(angle)
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```
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### Ks, Kv, Ka
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Divide the OLS terms by each other to obtain `Ks`, `Kv`, and `Ka`.
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```
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Ks = -γ/β
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Kv = -α/β
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Ka = 1/β
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```
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### Kg
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Take the sum of squares of the OLS terms containing the angle offset. The angle
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offset trig functions will form a trig identity that cancels out. Then, just
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solve for `Kg`.
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```
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δ²+ε² = (-Kg/Ka cos(offset))² + (Kg/Ka sin(offset))²
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δ²+ε² = (-Kg/Ka)² cos²(offset) + (Kg/Ka)² sin²(offset)
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δ²+ε² = (Kg/Ka)² cos²(offset) + (Kg/Ka)² sin²(offset)
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δ²+ε² = (Kg/Ka)² (cos²(offset) + sin²(offset))
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δ²+ε² = (Kg/Ka)² (1)
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δ²+ε² = (Kg/Ka)²
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√(δ²+ε²) = Kg/Ka
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√(δ²+ε²) = Kg β
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Kg = √(δ²+ε²)/β
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```
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As a sanity check, when the offset is zero, ε is zero and the equation for
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`Kg` simplifies to -δ/β, the equation previously used by SysId.
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### Angle offset
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Divide ε by δ, combine the trig functions into `tan(offset)`, then use `atan2()`
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to preserve the angle quadrant. Maintaining the proper negative signs in the
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numerator and denominator are important for obtaining the correct result.
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```
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δ = -Kg/Ka cos(offset)
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ε = Kg/Ka sin(offset)
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sin(offset)/-cos(offset) = ε/δ
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sin(offset)/cos(offset) = ε/-δ
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tan(offset) = ε/-δ
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offset = atan2(ε, -δ)
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```
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